-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

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 25.5: b) Find the expression for the magnitude of the electric field at P Start with and

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 sphere is a hyperbola