Two parallel-plate capacitors, identical except that one has twice the plate separation of the other, are charged by the Same voltage source. Which capacitor has a stronger electric field Between the plates? Which capacitor has a greater charge? Which has greater energy density? Explain your reasoning.

Energy density can be measured in energy per volume or per mass. The higher the energy density of a system or material, the greater the amount of energy it has stored.

Given

We are given two parallel-plate capacitors, identically except that one has twice the plate separation this means the next

Required

Which capacitor has a stronger electric field E, charge Q and energy density u

Explanation

The electric field depends on the separated distance between the two plates and it is given by

As shown by equation (1), the electric field is inversely proportional to the separated distance d as the distance increases the electric field decreases.

If we solve equation (1) to E1 and E2 we could get the ratio between the two electric fields as next

Therefore, the capacitor with the smaller separated distance has a greater electric field.

The distributed charge over the plates is related the the potential difference and the capacitance of the capacitor in the next form

But the capacitance is related to the distance between the two plates as shown by equation 24.2

Let us use this expression for C into equation (2), we will find that

As shown by this equation, the charge is inversely proportional to the distance d. If we solve equation (2) to Q1 and Q2 we could get the ratio between the two charges as next where the two capacitors have the same area

Therefore, the capacitor with the smaller separated distance has a greater charge.

The energy density depends on the electric field as shown by equation 23.11 in the form

As shown by this equation, the capacitor with a larger electric field has a larger energy density. The capacitor with the smallestseparateddistance has the largest electric field, hence has the largest energy density.

Therefore the The capacitor with a larger electric field has a larger energy density