Intuition for capacitors in seriesCapacitors in series?Capacitors in series: Why is the equivalent charge the same as the individual chargesCapacitors in a series circuit with dielectricCircuit with three capacitorsVoltage drop across capacitors in series, why?Relyance of plate distance on the capacitor equationUnsure of capacitor in series & in parallelEquivalent capacitance in seriesCapacitors in Series, which uses the most voltage?Infinite ladder of capacitors
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Intuition for capacitors in series
Capacitors in series?Capacitors in series: Why is the equivalent charge the same as the individual chargesCapacitors in a series circuit with dielectricCircuit with three capacitorsVoltage drop across capacitors in series, why?Relyance of plate distance on the capacitor equationUnsure of capacitor in series & in parallelEquivalent capacitance in seriesCapacitors in Series, which uses the most voltage?Infinite ladder of capacitors
.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;
$begingroup$
Can someone please explain, intuitively (without any formula, I understand the formulas), why the equivalent capacitance of capacitors in series is less than the any individual capacitor's capacitance?
PS: Let's take a simple case. Say we have 2 capacitors with Capacitance 2 (ignoring units), and we place them in series. A voltage $V$ develops in both, and a charge $+Q$ acculumates on one of their plates. Using the capacitance formulas, the equivalent capacitance is $1/2$ the original. Indeed we get $Q/2V$, where $Q/V$ is the original capacitance. But why? aren't we in total acculumating a charge of $2Q$ over a potential difference of $2V$? Why just $1Q$? (again, I'm speaking intuitively)
capacitance
$endgroup$
|
show 3 more comments
$begingroup$
Can someone please explain, intuitively (without any formula, I understand the formulas), why the equivalent capacitance of capacitors in series is less than the any individual capacitor's capacitance?
PS: Let's take a simple case. Say we have 2 capacitors with Capacitance 2 (ignoring units), and we place them in series. A voltage $V$ develops in both, and a charge $+Q$ acculumates on one of their plates. Using the capacitance formulas, the equivalent capacitance is $1/2$ the original. Indeed we get $Q/2V$, where $Q/V$ is the original capacitance. But why? aren't we in total acculumating a charge of $2Q$ over a potential difference of $2V$? Why just $1Q$? (again, I'm speaking intuitively)
capacitance
$endgroup$
$begingroup$
Is it intuitive for you that for capacitors in parallel, the equivalent capacitance is equal to the sum of all the capacitors?
$endgroup$
– PM 2Ring
8 hours ago
$begingroup$
Actually, I haven't thought about that case yet, because the book I'm using presents the series equations first. Are you suggesting that understanding the parallel case sheds light on this case?
$endgroup$
– user10796158
8 hours ago
$begingroup$
Well, for one thing the first capacitor has to charge and then discharge before the second capacitor can charge.
$endgroup$
– BillDOe
8 hours ago
1
$begingroup$
Not necessarily, but I think the parallel case is easier to understand than the series case, so to me it makes sense to consider it first.
$endgroup$
– PM 2Ring
8 hours ago
1
$begingroup$
@PM2Ring you're right, I just read that section, and it actually makes sense why you would add the capacitances. In a way, it's like you're increasing the area by adding in parallel for the same voltage drop. But the series case still doesn't quite make sense to me.
$endgroup$
– user10796158
8 hours ago
|
show 3 more comments
$begingroup$
Can someone please explain, intuitively (without any formula, I understand the formulas), why the equivalent capacitance of capacitors in series is less than the any individual capacitor's capacitance?
PS: Let's take a simple case. Say we have 2 capacitors with Capacitance 2 (ignoring units), and we place them in series. A voltage $V$ develops in both, and a charge $+Q$ acculumates on one of their plates. Using the capacitance formulas, the equivalent capacitance is $1/2$ the original. Indeed we get $Q/2V$, where $Q/V$ is the original capacitance. But why? aren't we in total acculumating a charge of $2Q$ over a potential difference of $2V$? Why just $1Q$? (again, I'm speaking intuitively)
capacitance
$endgroup$
Can someone please explain, intuitively (without any formula, I understand the formulas), why the equivalent capacitance of capacitors in series is less than the any individual capacitor's capacitance?
PS: Let's take a simple case. Say we have 2 capacitors with Capacitance 2 (ignoring units), and we place them in series. A voltage $V$ develops in both, and a charge $+Q$ acculumates on one of their plates. Using the capacitance formulas, the equivalent capacitance is $1/2$ the original. Indeed we get $Q/2V$, where $Q/V$ is the original capacitance. But why? aren't we in total acculumating a charge of $2Q$ over a potential difference of $2V$? Why just $1Q$? (again, I'm speaking intuitively)
capacitance
capacitance
edited 35 mins ago
Emilio Pisanty
88.7k23 gold badges225 silver badges458 bronze badges
88.7k23 gold badges225 silver badges458 bronze badges
asked 9 hours ago
user10796158user10796158
274 bronze badges
274 bronze badges
$begingroup$
Is it intuitive for you that for capacitors in parallel, the equivalent capacitance is equal to the sum of all the capacitors?
$endgroup$
– PM 2Ring
8 hours ago
$begingroup$
Actually, I haven't thought about that case yet, because the book I'm using presents the series equations first. Are you suggesting that understanding the parallel case sheds light on this case?
$endgroup$
– user10796158
8 hours ago
$begingroup$
Well, for one thing the first capacitor has to charge and then discharge before the second capacitor can charge.
$endgroup$
– BillDOe
8 hours ago
1
$begingroup$
Not necessarily, but I think the parallel case is easier to understand than the series case, so to me it makes sense to consider it first.
$endgroup$
– PM 2Ring
8 hours ago
1
$begingroup$
@PM2Ring you're right, I just read that section, and it actually makes sense why you would add the capacitances. In a way, it's like you're increasing the area by adding in parallel for the same voltage drop. But the series case still doesn't quite make sense to me.
$endgroup$
– user10796158
8 hours ago
|
show 3 more comments
$begingroup$
Is it intuitive for you that for capacitors in parallel, the equivalent capacitance is equal to the sum of all the capacitors?
$endgroup$
– PM 2Ring
8 hours ago
$begingroup$
Actually, I haven't thought about that case yet, because the book I'm using presents the series equations first. Are you suggesting that understanding the parallel case sheds light on this case?
$endgroup$
– user10796158
8 hours ago
$begingroup$
Well, for one thing the first capacitor has to charge and then discharge before the second capacitor can charge.
$endgroup$
– BillDOe
8 hours ago
1
$begingroup$
Not necessarily, but I think the parallel case is easier to understand than the series case, so to me it makes sense to consider it first.
$endgroup$
– PM 2Ring
8 hours ago
1
$begingroup$
@PM2Ring you're right, I just read that section, and it actually makes sense why you would add the capacitances. In a way, it's like you're increasing the area by adding in parallel for the same voltage drop. But the series case still doesn't quite make sense to me.
$endgroup$
– user10796158
8 hours ago
$begingroup$
Is it intuitive for you that for capacitors in parallel, the equivalent capacitance is equal to the sum of all the capacitors?
$endgroup$
– PM 2Ring
8 hours ago
$begingroup$
Is it intuitive for you that for capacitors in parallel, the equivalent capacitance is equal to the sum of all the capacitors?
$endgroup$
– PM 2Ring
8 hours ago
$begingroup$
Actually, I haven't thought about that case yet, because the book I'm using presents the series equations first. Are you suggesting that understanding the parallel case sheds light on this case?
$endgroup$
– user10796158
8 hours ago
$begingroup$
Actually, I haven't thought about that case yet, because the book I'm using presents the series equations first. Are you suggesting that understanding the parallel case sheds light on this case?
$endgroup$
– user10796158
8 hours ago
$begingroup$
Well, for one thing the first capacitor has to charge and then discharge before the second capacitor can charge.
$endgroup$
– BillDOe
8 hours ago
$begingroup$
Well, for one thing the first capacitor has to charge and then discharge before the second capacitor can charge.
$endgroup$
– BillDOe
8 hours ago
1
1
$begingroup$
Not necessarily, but I think the parallel case is easier to understand than the series case, so to me it makes sense to consider it first.
$endgroup$
– PM 2Ring
8 hours ago
$begingroup$
Not necessarily, but I think the parallel case is easier to understand than the series case, so to me it makes sense to consider it first.
$endgroup$
– PM 2Ring
8 hours ago
1
1
$begingroup$
@PM2Ring you're right, I just read that section, and it actually makes sense why you would add the capacitances. In a way, it's like you're increasing the area by adding in parallel for the same voltage drop. But the series case still doesn't quite make sense to me.
$endgroup$
– user10796158
8 hours ago
$begingroup$
@PM2Ring you're right, I just read that section, and it actually makes sense why you would add the capacitances. In a way, it's like you're increasing the area by adding in parallel for the same voltage drop. But the series case still doesn't quite make sense to me.
$endgroup$
– user10796158
8 hours ago
|
show 3 more comments
3 Answers
3
active
oldest
votes
$begingroup$
Can someone please explain, intuitively (without any formula, I
understand the formulas), why the equivalent capacitance of capacitors
in series is less than the any individual capacitor's capacitance?
I assume you know that the larger the capacitor plates are, the greater the capacitance, all other things being equal. Also I assume you know the greater the separation of the plates (the thicker the dielectric between the plates) the less the capacitance all other things being equal. Given these assumptions, consider the diagrams below.
The top diagram to the left shows two capacitors in parallel. It is equivalent to the digram to the top right. If two or more capacitors are connected in parallel, the overall effect is that of a single (equivalent) capacitor having a total plate area equal to the sum of the plate areas of the individual capacitors. Thus for parallel capacitors the equivalent capacitance is the sum of the capacitances.
The bottom diagram to the left shows two capacitors in series. It is equivalent to the diagram to the bottom right. If two or more capacitors are connected in series, the overall effect is that of a single (equivalent) capacitor having the sum total of the plate spacings of the individual capacitors. Thus for series capacitors the equivalent capacitor is less than the individual capacitors.
For reference the diagram includes an equation for capacitance based on the physical parameters of plate area $A$, plate separation $d$, and electrical permittivity $e$ of the dielectric.
Hope this helps.
$endgroup$
$begingroup$
Bob, thank you for your answer. This is starting to make sense. But do you mind elaborating a teeny bit more on why the total charge for the series case is Q not 2Q?
$endgroup$
– user10796158
7 hours ago
$begingroup$
Because it seems to me that the overall effect of having two plates in series each with charge Q would give 2Q
$endgroup$
– user10796158
7 hours ago
$begingroup$
@user10796158 Ok but I need some time to respond because I need to tend to something else at this time. Please stand by.
$endgroup$
– Bob D
7 hours ago
add a comment |
$begingroup$
The voltage change across a capacitor, or any region of space, is the negative path integral of field dot path. So if two capacitors are placed in series, you have twice the distance for the path integral, compared to one capacitor by itself. Now, since the distance is doubled, to get a final voltage equal to the battery that charges it, the electric field only needs to be 1/2 the electric field of the single capacitor. Since the electric field is proportional to the charge density, you only need 1/2 the charge density that would be on the single capacitor, so C = (Qsingle/2)/V.
$endgroup$
add a comment |
$begingroup$
why the equivalent capacitance of capacitors in series is less than
the any individual capacitor's capacitance?
To help build your intuition, place in series with another capacitor an ideal parallel plate capacitor (vacuum dielectric for simplicity), with plate spacing $d$. Now consider the limit as the distance $d$ goes to zero (assume zero initial charge).
Intuitively, the capacitance of the ideal parallel plate capacitor increases without bound and what's left in the limit is, effectively, an ideal short circuit when $d = 0$ (the two plates touch).
That is, the series combination of two capacitors has become a capacitor in series with an ideal short circuit. Clearly, the combined capacitance is just the capacitance of the remaining capacitor.
Now, hold the spacing $d$ constant and let the plate area $A$ go to zero instead. Intuitively, the capacitance of the ideal plate capacitor goes to zero and what's left in the limit is, effectively, an ideal open circuit.
That is, the series combination of two capacitors has become a capacitor in series with an ideal open circuit. Clearly, the combined capacitance is zero.
It follows that the total capacitance for two series capacitors with finite capacitance is less than the smaller of the two capacitances
$endgroup$
add a comment |
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3 Answers
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votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Can someone please explain, intuitively (without any formula, I
understand the formulas), why the equivalent capacitance of capacitors
in series is less than the any individual capacitor's capacitance?
I assume you know that the larger the capacitor plates are, the greater the capacitance, all other things being equal. Also I assume you know the greater the separation of the plates (the thicker the dielectric between the plates) the less the capacitance all other things being equal. Given these assumptions, consider the diagrams below.
The top diagram to the left shows two capacitors in parallel. It is equivalent to the digram to the top right. If two or more capacitors are connected in parallel, the overall effect is that of a single (equivalent) capacitor having a total plate area equal to the sum of the plate areas of the individual capacitors. Thus for parallel capacitors the equivalent capacitance is the sum of the capacitances.
The bottom diagram to the left shows two capacitors in series. It is equivalent to the diagram to the bottom right. If two or more capacitors are connected in series, the overall effect is that of a single (equivalent) capacitor having the sum total of the plate spacings of the individual capacitors. Thus for series capacitors the equivalent capacitor is less than the individual capacitors.
For reference the diagram includes an equation for capacitance based on the physical parameters of plate area $A$, plate separation $d$, and electrical permittivity $e$ of the dielectric.
Hope this helps.
$endgroup$
$begingroup$
Bob, thank you for your answer. This is starting to make sense. But do you mind elaborating a teeny bit more on why the total charge for the series case is Q not 2Q?
$endgroup$
– user10796158
7 hours ago
$begingroup$
Because it seems to me that the overall effect of having two plates in series each with charge Q would give 2Q
$endgroup$
– user10796158
7 hours ago
$begingroup$
@user10796158 Ok but I need some time to respond because I need to tend to something else at this time. Please stand by.
$endgroup$
– Bob D
7 hours ago
add a comment |
$begingroup$
Can someone please explain, intuitively (without any formula, I
understand the formulas), why the equivalent capacitance of capacitors
in series is less than the any individual capacitor's capacitance?
I assume you know that the larger the capacitor plates are, the greater the capacitance, all other things being equal. Also I assume you know the greater the separation of the plates (the thicker the dielectric between the plates) the less the capacitance all other things being equal. Given these assumptions, consider the diagrams below.
The top diagram to the left shows two capacitors in parallel. It is equivalent to the digram to the top right. If two or more capacitors are connected in parallel, the overall effect is that of a single (equivalent) capacitor having a total plate area equal to the sum of the plate areas of the individual capacitors. Thus for parallel capacitors the equivalent capacitance is the sum of the capacitances.
The bottom diagram to the left shows two capacitors in series. It is equivalent to the diagram to the bottom right. If two or more capacitors are connected in series, the overall effect is that of a single (equivalent) capacitor having the sum total of the plate spacings of the individual capacitors. Thus for series capacitors the equivalent capacitor is less than the individual capacitors.
For reference the diagram includes an equation for capacitance based on the physical parameters of plate area $A$, plate separation $d$, and electrical permittivity $e$ of the dielectric.
Hope this helps.
$endgroup$
$begingroup$
Bob, thank you for your answer. This is starting to make sense. But do you mind elaborating a teeny bit more on why the total charge for the series case is Q not 2Q?
$endgroup$
– user10796158
7 hours ago
$begingroup$
Because it seems to me that the overall effect of having two plates in series each with charge Q would give 2Q
$endgroup$
– user10796158
7 hours ago
$begingroup$
@user10796158 Ok but I need some time to respond because I need to tend to something else at this time. Please stand by.
$endgroup$
– Bob D
7 hours ago
add a comment |
$begingroup$
Can someone please explain, intuitively (without any formula, I
understand the formulas), why the equivalent capacitance of capacitors
in series is less than the any individual capacitor's capacitance?
I assume you know that the larger the capacitor plates are, the greater the capacitance, all other things being equal. Also I assume you know the greater the separation of the plates (the thicker the dielectric between the plates) the less the capacitance all other things being equal. Given these assumptions, consider the diagrams below.
The top diagram to the left shows two capacitors in parallel. It is equivalent to the digram to the top right. If two or more capacitors are connected in parallel, the overall effect is that of a single (equivalent) capacitor having a total plate area equal to the sum of the plate areas of the individual capacitors. Thus for parallel capacitors the equivalent capacitance is the sum of the capacitances.
The bottom diagram to the left shows two capacitors in series. It is equivalent to the diagram to the bottom right. If two or more capacitors are connected in series, the overall effect is that of a single (equivalent) capacitor having the sum total of the plate spacings of the individual capacitors. Thus for series capacitors the equivalent capacitor is less than the individual capacitors.
For reference the diagram includes an equation for capacitance based on the physical parameters of plate area $A$, plate separation $d$, and electrical permittivity $e$ of the dielectric.
Hope this helps.
$endgroup$
Can someone please explain, intuitively (without any formula, I
understand the formulas), why the equivalent capacitance of capacitors
in series is less than the any individual capacitor's capacitance?
I assume you know that the larger the capacitor plates are, the greater the capacitance, all other things being equal. Also I assume you know the greater the separation of the plates (the thicker the dielectric between the plates) the less the capacitance all other things being equal. Given these assumptions, consider the diagrams below.
The top diagram to the left shows two capacitors in parallel. It is equivalent to the digram to the top right. If two or more capacitors are connected in parallel, the overall effect is that of a single (equivalent) capacitor having a total plate area equal to the sum of the plate areas of the individual capacitors. Thus for parallel capacitors the equivalent capacitance is the sum of the capacitances.
The bottom diagram to the left shows two capacitors in series. It is equivalent to the diagram to the bottom right. If two or more capacitors are connected in series, the overall effect is that of a single (equivalent) capacitor having the sum total of the plate spacings of the individual capacitors. Thus for series capacitors the equivalent capacitor is less than the individual capacitors.
For reference the diagram includes an equation for capacitance based on the physical parameters of plate area $A$, plate separation $d$, and electrical permittivity $e$ of the dielectric.
Hope this helps.
edited 7 hours ago
answered 8 hours ago
Bob DBob D
8,7603 gold badges7 silver badges29 bronze badges
8,7603 gold badges7 silver badges29 bronze badges
$begingroup$
Bob, thank you for your answer. This is starting to make sense. But do you mind elaborating a teeny bit more on why the total charge for the series case is Q not 2Q?
$endgroup$
– user10796158
7 hours ago
$begingroup$
Because it seems to me that the overall effect of having two plates in series each with charge Q would give 2Q
$endgroup$
– user10796158
7 hours ago
$begingroup$
@user10796158 Ok but I need some time to respond because I need to tend to something else at this time. Please stand by.
$endgroup$
– Bob D
7 hours ago
add a comment |
$begingroup$
Bob, thank you for your answer. This is starting to make sense. But do you mind elaborating a teeny bit more on why the total charge for the series case is Q not 2Q?
$endgroup$
– user10796158
7 hours ago
$begingroup$
Because it seems to me that the overall effect of having two plates in series each with charge Q would give 2Q
$endgroup$
– user10796158
7 hours ago
$begingroup$
@user10796158 Ok but I need some time to respond because I need to tend to something else at this time. Please stand by.
$endgroup$
– Bob D
7 hours ago
$begingroup$
Bob, thank you for your answer. This is starting to make sense. But do you mind elaborating a teeny bit more on why the total charge for the series case is Q not 2Q?
$endgroup$
– user10796158
7 hours ago
$begingroup$
Bob, thank you for your answer. This is starting to make sense. But do you mind elaborating a teeny bit more on why the total charge for the series case is Q not 2Q?
$endgroup$
– user10796158
7 hours ago
$begingroup$
Because it seems to me that the overall effect of having two plates in series each with charge Q would give 2Q
$endgroup$
– user10796158
7 hours ago
$begingroup$
Because it seems to me that the overall effect of having two plates in series each with charge Q would give 2Q
$endgroup$
– user10796158
7 hours ago
$begingroup$
@user10796158 Ok but I need some time to respond because I need to tend to something else at this time. Please stand by.
$endgroup$
– Bob D
7 hours ago
$begingroup$
@user10796158 Ok but I need some time to respond because I need to tend to something else at this time. Please stand by.
$endgroup$
– Bob D
7 hours ago
add a comment |
$begingroup$
The voltage change across a capacitor, or any region of space, is the negative path integral of field dot path. So if two capacitors are placed in series, you have twice the distance for the path integral, compared to one capacitor by itself. Now, since the distance is doubled, to get a final voltage equal to the battery that charges it, the electric field only needs to be 1/2 the electric field of the single capacitor. Since the electric field is proportional to the charge density, you only need 1/2 the charge density that would be on the single capacitor, so C = (Qsingle/2)/V.
$endgroup$
add a comment |
$begingroup$
The voltage change across a capacitor, or any region of space, is the negative path integral of field dot path. So if two capacitors are placed in series, you have twice the distance for the path integral, compared to one capacitor by itself. Now, since the distance is doubled, to get a final voltage equal to the battery that charges it, the electric field only needs to be 1/2 the electric field of the single capacitor. Since the electric field is proportional to the charge density, you only need 1/2 the charge density that would be on the single capacitor, so C = (Qsingle/2)/V.
$endgroup$
add a comment |
$begingroup$
The voltage change across a capacitor, or any region of space, is the negative path integral of field dot path. So if two capacitors are placed in series, you have twice the distance for the path integral, compared to one capacitor by itself. Now, since the distance is doubled, to get a final voltage equal to the battery that charges it, the electric field only needs to be 1/2 the electric field of the single capacitor. Since the electric field is proportional to the charge density, you only need 1/2 the charge density that would be on the single capacitor, so C = (Qsingle/2)/V.
$endgroup$
The voltage change across a capacitor, or any region of space, is the negative path integral of field dot path. So if two capacitors are placed in series, you have twice the distance for the path integral, compared to one capacitor by itself. Now, since the distance is doubled, to get a final voltage equal to the battery that charges it, the electric field only needs to be 1/2 the electric field of the single capacitor. Since the electric field is proportional to the charge density, you only need 1/2 the charge density that would be on the single capacitor, so C = (Qsingle/2)/V.
answered 6 hours ago
lamplamplamplamp
5541 gold badge5 silver badges19 bronze badges
5541 gold badge5 silver badges19 bronze badges
add a comment |
add a comment |
$begingroup$
why the equivalent capacitance of capacitors in series is less than
the any individual capacitor's capacitance?
To help build your intuition, place in series with another capacitor an ideal parallel plate capacitor (vacuum dielectric for simplicity), with plate spacing $d$. Now consider the limit as the distance $d$ goes to zero (assume zero initial charge).
Intuitively, the capacitance of the ideal parallel plate capacitor increases without bound and what's left in the limit is, effectively, an ideal short circuit when $d = 0$ (the two plates touch).
That is, the series combination of two capacitors has become a capacitor in series with an ideal short circuit. Clearly, the combined capacitance is just the capacitance of the remaining capacitor.
Now, hold the spacing $d$ constant and let the plate area $A$ go to zero instead. Intuitively, the capacitance of the ideal plate capacitor goes to zero and what's left in the limit is, effectively, an ideal open circuit.
That is, the series combination of two capacitors has become a capacitor in series with an ideal open circuit. Clearly, the combined capacitance is zero.
It follows that the total capacitance for two series capacitors with finite capacitance is less than the smaller of the two capacitances
$endgroup$
add a comment |
$begingroup$
why the equivalent capacitance of capacitors in series is less than
the any individual capacitor's capacitance?
To help build your intuition, place in series with another capacitor an ideal parallel plate capacitor (vacuum dielectric for simplicity), with plate spacing $d$. Now consider the limit as the distance $d$ goes to zero (assume zero initial charge).
Intuitively, the capacitance of the ideal parallel plate capacitor increases without bound and what's left in the limit is, effectively, an ideal short circuit when $d = 0$ (the two plates touch).
That is, the series combination of two capacitors has become a capacitor in series with an ideal short circuit. Clearly, the combined capacitance is just the capacitance of the remaining capacitor.
Now, hold the spacing $d$ constant and let the plate area $A$ go to zero instead. Intuitively, the capacitance of the ideal plate capacitor goes to zero and what's left in the limit is, effectively, an ideal open circuit.
That is, the series combination of two capacitors has become a capacitor in series with an ideal open circuit. Clearly, the combined capacitance is zero.
It follows that the total capacitance for two series capacitors with finite capacitance is less than the smaller of the two capacitances
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add a comment |
$begingroup$
why the equivalent capacitance of capacitors in series is less than
the any individual capacitor's capacitance?
To help build your intuition, place in series with another capacitor an ideal parallel plate capacitor (vacuum dielectric for simplicity), with plate spacing $d$. Now consider the limit as the distance $d$ goes to zero (assume zero initial charge).
Intuitively, the capacitance of the ideal parallel plate capacitor increases without bound and what's left in the limit is, effectively, an ideal short circuit when $d = 0$ (the two plates touch).
That is, the series combination of two capacitors has become a capacitor in series with an ideal short circuit. Clearly, the combined capacitance is just the capacitance of the remaining capacitor.
Now, hold the spacing $d$ constant and let the plate area $A$ go to zero instead. Intuitively, the capacitance of the ideal plate capacitor goes to zero and what's left in the limit is, effectively, an ideal open circuit.
That is, the series combination of two capacitors has become a capacitor in series with an ideal open circuit. Clearly, the combined capacitance is zero.
It follows that the total capacitance for two series capacitors with finite capacitance is less than the smaller of the two capacitances
$endgroup$
why the equivalent capacitance of capacitors in series is less than
the any individual capacitor's capacitance?
To help build your intuition, place in series with another capacitor an ideal parallel plate capacitor (vacuum dielectric for simplicity), with plate spacing $d$. Now consider the limit as the distance $d$ goes to zero (assume zero initial charge).
Intuitively, the capacitance of the ideal parallel plate capacitor increases without bound and what's left in the limit is, effectively, an ideal short circuit when $d = 0$ (the two plates touch).
That is, the series combination of two capacitors has become a capacitor in series with an ideal short circuit. Clearly, the combined capacitance is just the capacitance of the remaining capacitor.
Now, hold the spacing $d$ constant and let the plate area $A$ go to zero instead. Intuitively, the capacitance of the ideal plate capacitor goes to zero and what's left in the limit is, effectively, an ideal open circuit.
That is, the series combination of two capacitors has become a capacitor in series with an ideal open circuit. Clearly, the combined capacitance is zero.
It follows that the total capacitance for two series capacitors with finite capacitance is less than the smaller of the two capacitances
answered 1 hour ago
Alfred CentauriAlfred Centauri
49.4k3 gold badges51 silver badges155 bronze badges
49.4k3 gold badges51 silver badges155 bronze badges
add a comment |
add a comment |
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$begingroup$
Is it intuitive for you that for capacitors in parallel, the equivalent capacitance is equal to the sum of all the capacitors?
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– PM 2Ring
8 hours ago
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Actually, I haven't thought about that case yet, because the book I'm using presents the series equations first. Are you suggesting that understanding the parallel case sheds light on this case?
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– user10796158
8 hours ago
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Well, for one thing the first capacitor has to charge and then discharge before the second capacitor can charge.
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– BillDOe
8 hours ago
1
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Not necessarily, but I think the parallel case is easier to understand than the series case, so to me it makes sense to consider it first.
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– PM 2Ring
8 hours ago
1
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@PM2Ring you're right, I just read that section, and it actually makes sense why you would add the capacitances. In a way, it's like you're increasing the area by adding in parallel for the same voltage drop. But the series case still doesn't quite make sense to me.
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– user10796158
8 hours ago