Two capacitors, C1 and C2, are connected to a battery whose...

Two capacitors, $C_{1}$ and $C_{2},$ are connected to a battery whose voltage is $V$. Recall from Section 19.5 that the electrical energy stored by each capacitor is $\frac{1}{2} C_{1} V_{1}^{2}$ and $\frac{1}{2} C_{2} V_{2}^{2},$ where $V_{1}$ and $V_{2}$ are, respectively, the voltages across $C_{1}$ and $C_{2}$. If the capacitors are connected in series, is the total energy stored by them greater than, less than, or equal to the total energy stored when they are connected in parallel? Justify your answer. Problem The battery voltage is $V=60.0 \mathrm{~V},$ and the capacitances are $C_{1}=2.00 \mu \mathrm{F}$ and $C_{2}=4.00 \mu \mathrm{F} .$ Determine the total energy stored by the two capacitors when they are wired (a) in parallel and (b) in series. Check to make sure that your answer is consistent with your answer to the Concept Question.

Key Concepts

Energy Stored in Capacitors

The energy stored in a capacitor is calculated using the formula E = 1/2 × C × V², where C is the capacitance and V is the voltage across the capacitor. This relationship shows that the stored energy is directly proportional to both the capacitance and the square of the voltage. How the capacitors are arranged (in series or parallel) directly influences the effective capacitance and the distribution of voltage, thereby affecting the total energy stored.

Capacitors in Series

In a series configuration, capacitors are connected end-to-end so that they share the same charge, but the voltage across each capacitor divides in inverse proportion to their capacitances. The equivalent capacitance for capacitors in series is less than the smallest individual capacitance, leading to a lower overall energy storage compared to the parallel arrangement when connected to the same voltage source.

Capacitance

Capacitance is a measure of a capacitor’s ability to store charge per unit voltage. It reflects how much electric charge can be stored on the capacitor for a given potential difference and is measured in farads. In general, a higher capacitance means that the capacitor can store more charge, which is crucial in various electrical circuits for energy storage and filtering applications.

Capacitors in Parallel

When capacitors are connected in parallel, they all share the same voltage as the source of power. The total or equivalent capacitance is simply the sum of the individual capacitances, which allows the circuit to store more charge overall. This configuration maximizes the energy storage since each capacitor operates at the full voltage of the battery.

Transcript

00:01 Here we have a question about the energy storage capacitors.

00:04 So we're given a formula for energy storing capacitors, and then we're told that we have two different setups that we can use with two different capacitors, with different capacitance.

00:17 Both have the same emf for the battery.

00:21 And first we're just asked qualitatively which arrangement can store more energy.

00:29 So we want to find some inequality which relates the energy in a series combination to the energy in parallel combination.

00:41 This equal sign with a question mark over it means we haven't yet determined what this inequality is.

00:47 But it doesn't mean it's these two are equal.

00:50 So let's find out how they relate.

00:52 I individually finding what their energies are in terms of c1, c2, and b.

00:56 So the total energy in series is going to be the energy cross -capaster 1 plus energy on capacitor 2.

01:07 So that's going to be 1 -5 -c -1 -b -1 squared plus 1 -5 -c2, b -2 squared, where b -1 is the voltage across capacitor 1 and b -2's voltage across -capaster 2.

01:22 All right, so now let's write it out for the energy in the parallel combination.

01:28 So this is going to be 1 half c1 v squared.

01:33 Notice not v1, just v, plus 1⁄2 v squared.

01:40 All right, the reason it's just v is because in the parallel combination, both of these loops have to have a voltage drop of v across the capacitor.

01:50 Since krikos loop rule says that when we go around the loop, we have to have a net changing voltage at zero.

01:59 So we know that v is the sum of v1 plus v2.

02:05 Therefore, v is greater than both v1 and v2.

02:09 So since these are greater than these, then as a whole, the energy in parallel must be greater than the energy in series.

02:18 So this inequality is less than sign.

02:23 I suppose it can be less than equal to in a case where one of the capacters capacitance was nearly zero, but we're going to leave that case out.

02:36 So it's just a less than fine.

02:39 Ok, so now in part two of the question, they tell us that the exact value of the voltage in the batteries is 60 volts.

02:50 And they tell us that c1 is me, switch back to believe, to be consistent.

02:57 C1 has a capacitance of two microfarid and so that's a bad a microfarid and c2 has a best in of four microperds.

03:12 So now they want us to find the energy in the serious combination and in the parallel combination.

03:20 And then we can compare those and see if the inequality holds.

03:23 All right, so let's start with the energy in the parallel combination.

03:31 So we establish that this is equal to one half, c1, b squared, plus one half c2 v squared.

03:39 Now this isn't hard at all.

03:41 We can just plug in c1, c2, and v.

03:44 We have all the numbers we need.

03:46 So we'll just plug those in and we find that the energy in the parallel combination is 6 times 10 to minus 10 joules.

03:55 So that's pretty easy to calculate.

03:57 What makes the energy in series harder to calculate is that the voltage across each capacitor is not as obvious as just b...

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