-Electric Potential due to Continuous Charge Distributions

AP Physics C

Mrs. Coyle

Electric Potential –What we used so far! Electric Potential

Potential Difference

Potential for a point charge

Potential for multiple point charges

o

UV

q

B

Ao

UV d

q

E s

e

qV k

r

ie

i i

qV k

r

Remember: V is a scalar quantity

Keep the signs of the charges in the equations, so V is positive for positive charges.

You need a reference V because it is changes in electric potential that are significant. When dealing with point charges and charge distributions the reference is V=0 when r

Electric Potential Due to a Continuous Charge Distribution

How would you calculate the V at point P?

Two Ways to Calculate Electric Potential Due to a Continuous Charge Distribution

It can be calculated in two ways: Method 1: Divide the

surface into infinitesimal elements dq

Method 2:If E is known (from Gauss’s Law) B

Ao

UV d

q

E s

e

dqV k

r

/ oE dA q

Method 1 Consider an infinitesimal

charge element dq and treat it as a point charge

The potential at point P due to dq

e

dqdV k

r

Method 1 Cont’d For the total potential, integrate to include the

contributions from all the dq elements

Note: reference of V = 0 is when P is an infinite distance from the charge distribution.

e

dqV k

r

Ex 25.5 : a) V at a point on the perpendicular central axis of a Uniformly Charged Ring

Assume that the total

charge of the ring is Q.

Show that:

2 2

ee

k QdqV k

r x a

Ex 25.5: b) Find the expression for the magnitude of the electric field at P

Start with

and

2 2

ek QV

x a

xdV

Edx

x 2 2 3/ 2Ans: E

( )

Note that at the center of the ring E=0.

How else had we calculated this result?

kQx

x a

Ex 25.6: Find a)V and b) E at a point along the central perpendicular axis of a Uniformly Charged Disk

Assume radius a and surface charge density of σ. Assume that a disk is a series of many rings with width dr.

12 2 2Ans.: 2 eV πk σ x a x

Ex 25.6: Find a)V and b) E at a point along the central perpendicular axis of a Uniformly Charged Disk

12 2 2Start with 2 eV πk σ x a x

2 2Ans: 2 (1 )x

xE k

x a

Ex25.7: Find V at a point P a distance a from a Finite Line of Charge Assume the total charge of

the rod is Q, length l and a linear charge density of λ.

Hint:

2 2

Ans: lnek Q aV

a

2 2

2 2ln( )

dxx x a

x a

Method 2 for Calculating V for a Continuous Charge Distribution:

If E is known (from Gauss’s Law)

Then use:

/ oE dA q

B

Ao

UV d

q

E s

Ex 25.8: Find V for a Uniformly Charged Sphere (Hint: Use Gauss’s Law to find E) Assume a solid

insulating sphere of radius R and total charge Q

For r > R,

: e

QAns V k

r

Ex 25.8: Find V for a Uniformly Charged Sphere A solid sphere of

radius R and total charge Q

For r < R,

2 23

:2

eD C

k QAns V V R r

R

Ex 25.8:V for a Uniformly Charged Sphere, Graph The curve for inside

the sphere is parabolic

The curve for outside the sphere is a hyperbola