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-Electric Potential due to Continuous Charge Distributions

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Electric Potential –What we used so far!

- Electric Potential
- Potential Difference
- Potential for a point charge
- Potential for multiple point charges

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)

Method 1

- Consider an infinitesimal charge element dq and treat it as a point charge
- The potential at point P due to dq

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.

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:

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.

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

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:

Method 2 for Calculating V for a Continuous Charge Distribution:

- If E is known (from Gauss’s Law)
- Then use:

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,

Ex 25.8: Find V for a Uniformly Charged Sphere

- A solid sphere of radius R and total charge Q
- For 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 sphereis a hyperbola

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