If an uncharged parallel-plate capacitor (capacitance C ) is...

If an uncharged parallel-plate capacitor (capacitance $C$ ) is connected to a battery, one plate becomes negatively charged as electrons move to the plate face (area $A$ ). In Fig. $25-32$, the depth $d$ from which the electrons come in the plate in a particular capacitor is plotted against a range of values for the potential difference $V$ of the battery. The density of conduction electrons in the copper plates is $8.49 \times 10^{28}$ electrons $/ \mathrm{m}^{3}$. The vertical scale is set by $d_{s}=2.00 \mathrm{pm}$, and the horizontal scale is set by $V_{s}=20.0 \mathrm{~V}$. What is the ratio $C / A$ ?

Key Concepts

Relation Between Charge, Voltage, and Electron Displacement

When a capacitor is connected to a battery, the applied voltage causes a charge Q to accumulate on the plates, which, on a microscopic level, corresponds to a small net displacement of electrons within the metal over a depth. This microscopic displacement multiplied by the number density of conduction electrons and the elementary charge yields the total charge on the plate. Relating Q = C·V to this microscopic picture allows one to derive expressions for capacitance, such as C/A, based on fundamental constants and measurable displacement.

Microscopic Charge Distribution and Electron Displacement

In conductors, the free electrons are not rigidly fixed; a small displacement of these conduction electrons under an applied voltage can cause a net charge accumulation on the surfaces of the metal. The amount of charge displaced from a region in the metal is determined by the conduction electron density and the displacement depth. This microscopic picture helps bridge the gap between the atomic-scale movement of electrons and the macroscopic electrical properties such as capacitance.

Capacitance per Unit Area

This concept involves evaluating the ability of a capacitor to store charge relative to its plate area. In a parallel?plate capacitor, the capacitance is directly proportional to the area and inversely proportional to the separation between the plates. When considering the capacitance per unit area, one looks at the ratio C/A, which, especially in vacuum or with a dielectric, relates to the permittivity of free space and geometric factors. It is a useful measure for comparing capacitors independent of their absolute sizes.

Transcript

00:01 In this question we have been given an uncharged parallel plate capacitor which has the capacitance as c and it is given that there is this graph.

00:10 D here denotes the depth whereas v denotes the potential drop and the density of conduction of electrons in these copper plates of the capacitor is given as so let's write down the important values.

00:23 So n here which is the number density of electrons is given to us as 8 .4.

00:30 49 into 10 raised to the power 28.

00:33 These are electrons per meter cube because this is in volume number density.

00:40 And they've also given us the vertical scales which is set to that means the depth here, here which is ds is set to it's set to two picometer which is 2 pico meters which is 2 into 10 raised to the power minus 12 meters.

00:55 And they've also given us the horizontal scale that means in the vertical, the horizontal axis which is the x -axis the value for voltage let's write it down as vs is set as 20 volts we have to find out the ratio so let us write down what we have to find out we have to find out the ratio of capacitance and to the area so this is what we have to find out let us write down some important formulas over here which we'll be using so the first thing that we need to know is that q is equal to n e so this is call the quantization of charge.

01:32 So it's quantization of charge.

01:37 So this is the formula which is used.

01:40 So in this formula, we actually say that charge can only go in terms of, you know, multiple of integrals.

01:46 So this value n can be one, two, three, or any integral only.

01:51 So let us write this formula in another way.

01:55 So we know n number density is a number of electrons per unit volume.

02:00 So here these are the total electrons.

02:02 So this becomes small n which let's write this as capital n to differentiate.

02:09 So these are the total number of electrons whereas this is the number density.

02:13 So this is small n, a into d.

02:16 So for volume you multiply depth with the area and this entire thing multiplied with e.

02:22 So this is a formula that we have for the charge...

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