Capacitor A and capacitor B both have the same voltage...

Capacitor A and capacitor B both have the same voltage across their plates. However, the energy of capacitor A can melt $m$ kilograms of ice at $0{ }^{\circ} \mathrm{C}$, while the energy of capacitor $\mathrm{B}$ can boil away the same amount of water at $100^{\circ} \mathrm{C}$. (a) Which requires more energy, melting the ice or boiling the water? (b) Which capacitor has the greater capacitance? Explain your answers. The capacitance of capacitor A is $9.3 \mu \mathrm{F}$. What is the capacitance of capacitor B? Be sure that your answer is consistent with your answers to the Concept Questions.

Capacitor A and capacitor B both have the same voltage across
their plates. However, the energy of capacitor A can melt $m$ kilograms of ice at $0{ }^{\circ} \mathrm{C}$, while the energy of capacitor $\mathrm{B}$ can boil away the same amount of water at $100^{\circ} \mathrm{C}$. (a) Which requires more energy, melting the ice or boiling the water? (b) Which capacitor has the greater capacitance? Explain your answers.
The capacitance of capacitor A is $9.3 \mu \mathrm{F}$. What is the capacitance of capacitor B? Be sure that your answer is consistent with your answers to the Concept Questions.

Key Concepts

Capacitor Energy Storage

This concept revolves around the ability of a capacitor to store electrical energy, which is described by the equation E = ½CV². Here, E is the stored energy, C is the capacitance, and V is the voltage across the capacitor. This relationship shows that, at a constant voltage, the stored energy is directly proportional to the capacitance, making it possible to compare energy contents solely based on capacitance when the voltage is fixed.

Latent Heat

Latent heat is the energy required to change the phase of a substance without changing its temperature. Two specific latent heat values are of interest: the latent heat of fusion, which is the energy needed to melt a solid into a liquid, and the latent heat of vaporization, which is the energy required to turn a liquid into a gas. Typically, the latent heat of vaporization is significantly larger, meaning boiling requires more energy than melting, even for the same mass of substance.

Relationship Between Energy, Capacitance, and Process Requirements

By combining the concept of capacitor energy storage with the latent heat values, one can deduce that if one process (such as boiling water) requires more energy than another process (melting ice), then at a fixed voltage the capacitor providing energy for the higher-energy process must have a greater capacitance. This relationship is critical for comparing different capacitors based on the energy demands of the phase changes they are used to drive.

Transcript

00:01 In this problem, we are given two capacitors a and b, and we are also given information about their energy content.

00:08 Technically, what the energy is stored in the electric field.

00:14 We also are told that the potential across each is the same.

00:19 So, v, a, v, b, just v.

00:26 And let's remember the formula for energy stored in electric field.

00:32 One half c v squared so seeing that the v is going to be the same, the energy between the two is the difference is because it's going to be the difference in the capacitance.

00:49 Now in part a, we're told that capacitor a can melt m kilograms of ice at zero degrees celsius.

00:58 So that means that energy a is m times lf where lf is a latent heat fusion.

01:14 This is 33 .5 times 10 to 4 joules per kilogram.

01:26 And then we are told that b can vaporize m kilograms of water at 100 degrees celsius.

01:39 So now we use the latent heat of vaporization.

01:44 And that is 22 .6 times 10 to the 5th.

01:53 Joules per kilogram and since lv as we can see it's greater than lf that means the vaporization the boiling takes more energy so that tells us that the energy the capacitor b has the largest energy content to it so that was step a just to really determine which of the two processes is requires more energy and now it's asking part b, which has then the largest capacitance? well, as i said, we can see that energy b is larger, greater than energy a implies.

03:08 And you can, as we mentioned before, the v is the same.

03:12 So everything, the only difference is in the capacitance...

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