Part A To be able to calculate the energy of a charged capacitor and to understa

Question

Part A To be able to calculate the energy of a charged capacitor and to understand the concept of energy associated with an electric Find the energy U of the capacitor in terms of C and Q by using the definiion of capacitance and the fomula for the enengy in a capacitor Express your answer in terms of C and Q. field The energy of a charged capacitor is given by U-QV/2 where Q is the charge of the capacitor and V is the potential d flerence across the capacilor. The energy of a charged can be described as the energy associated with the electric field created inside the capacitor In this problem, you will derive two more formulas for the energy Submit My Answers Give Up for the energy field. It will be useful to recall he d C=Q/V.and the formula for te Part B C=eoA/d.where Aisthe area of each ofthe plates anddis the plate separation As usual, Co is the permitivity of free space. This question will be shown after you complete previous question/s)

Explanation / Answer

A)

U = QV/2 .............. given in the problem   ............(1)

capacitance: C = Q/V    by defination of capacitance

rearranging,

V = Q/C,

Q = CV

put value of Q in equation (1)

U = QV/2

U = ((CV) * V) / 2

U = (CV2) / 2

C)

Initial: U = (CV2) / 2

Final: ((C / 2)V2) / 2

Divide final (U) by initial (U0). V2 cancels:

U/U0 = [(C / 2) / 2] / [C / 2]

U/U0 = (C/4) / (C/2)

U/U0 = 0.5

D)

Initial: U = Q2 / (2C)

Final: U = Q2 / (2(C / 2))

Divide final (U) by initial (U0). Q2 cancels:

U/U0 = [2C] / [(2(C / 2))]

U/U0 = 2C / C

U/U0 = 2

E)

C = 0A/d     ............(given)

U = (CV2) / 2

Substitute in for C:

U = (0A/d) * V2 / 2

U = (0AV2) / 2d

F)

U = (0AV2) / 2d

V = Ed, so V2 = E2 * d2

Substitute in for V

U = (0A*(E2*d2)) / 2d

Simplify,

U = (0AdE2) / 2

G)

energy density is u = U / V

U = (0AdE2) / 2

The Ad term is equivalent to the volume of a symmetrical box

A is the area of a base, d is the distance between the bases

So divide U by the volume (Ad)

U = ((0AdE2) / 2) / (Ad)

Simplify,

U = (0E2) / 2

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