Electric Potential of Uniform Charge Distributions

1. Electric Potential of Uniform Charge Distributions AP Physics C Montwood High School R. Casao

2. Electric Potential Due to a Charged Rod on the Axis of the Rod

3. Electric Potential Due to a Charged Rod on the Axis of the Rod • Divide the length of the rod L into small elements of length dx. • This also divides the total charge Q on the rod into small elements of charge dq. • Over the length L of the rod, the charge on each small piece of length dx is dq. • The charge density on the rod is:

4. Electric Potential Due to a Charged Rod on the Axis of the Rod • Because the charge is uniformly distributed over the length of the rod, we can set up a proportion relating the total charge per unit length to the charge per unit length for each small piece of the rod. • Each piece of length dx contains a charge dq and can be considered to be a point charge.

5. Electric Potential Due to a Charged Rod on the Axis of the Rod • Each element of charge dq contributes to the net electric potential V at point P1. • Since the charge on the rod is positive, the net electric potential V at point P1 is positive. • The distance from point P1 to each piece dx is x. • Electric potential equation for a point charge:

6. Electric Potential Due to a Charged Rod on the Axis of the Rod • The electric potential contribution from each element of charge dq is designated dV. • To determine the total electric potential V at point P1, add the electric potential contribution from each element of charge dq from the left end of the rod to the right end of the rod. • Integrate from d at the left end of the rod to (d + L) at the right end of the rod. In other words, the electric field contributions begin at d and end at (d + L). • This means that dq needs to be expressed in terms of dx because we will integrate along the x-axis from d to (d + L).

7. Electric Potential Due to a Charged Rod on the Axis of the Rod • For each point charge dq:

8. Electric Potential Due to a Charged Rod on the Axis of the Rod • Integrate both sides of the equation from the left end of the rod at x = d to the right end of the rod at d + L. • On the left side of the equation: the sum of the contributions to the electric potential dV at point P1 is the electric potential V.

9. Electric Potential Due to a Charged Rod on the Axis of the Rod • From a table of integrals (or your TI – 89): • Completing the integral:

10. Electric Potential Due to a Charged Rod on the Axis of the Rod • If Q is positive, l will be positive and V will be positive. • If Q is negative, l will be negative and V will be negative.

11. Electric Potential Off the Axis of a Finite Line of Charge

12. Electric Potential Off the Axis of a Finite Line of Charge • Divide the length of the rod L into small elements of length dx. • This also divides the total charge Q on the rod into small elements of charge dq. • Over the length L of the rod, the charge on each small element of length dx is dq. • The charge density on the rod is:

13. Electric Potential Off the Axis of a Finite Line of Charge • Because the charge is uniformly distributed over the length of the rod, we can set up a proportion relating the total charge per unit length to the charge per unit length for each small piece of the rod. • Each element of length dx contains a charge dq that can be considered to be a point charge. • Each element of charge dq contributes to the net electric potential V at point P2.

14. Electric Potential Off the Axis of a Finite Line of Charge • Electric potential equation for a point charge: • The electric potential contribution from each element of charge dq is designated dV:

15. Electric Potential Off the Axis of a Finite Line of Charge • At point P2, the net electric potential V will be the sum of the electric potential contribution dV of each element of charge dq from the left end of the rod (x = 0) to the right end of the rod (x = L).

16. Electric Potential Off the Axis of a Finite Line of Charge • For each element of charge dq from x = 0 to x = L, the values of V, r, and q all change as x changes. The relationships between these variables must be determined. • Because electric potential is a scalar quantity and not a vector quantity, we do not have to worry about the angle q for the direction. We only have to concern ourselves with the distance r from element of charge dq to point P2.

18. Electric Potential Off the Axis of a Finite Line of Charge • Integrate from 0 at the left end of the rod to L at the right end of the rod. In other words, the electric potential contributions begin at 0 and end at L. • This means that dq needs to be expressed in terms of dx because we will integrate along the x axis as the distance x changes from 0 to L.

19. Electric Potential Off the Axis of a Finite Line of Charge • Integrate both sides of the equation from 0 to L:

20. Electric Potential Off the Axis of a Finite Line of Charge • On the left side of the integral: adding each contribution dV from the left end of the rod to the right end of the rod gives us the total potential V: • Pull the constants k and l out in front of the integral sign and integrate. The y2 term cannot be removed from the integration because it is being added to the x2 term.

21. Electric Potential Off the Axis of a Finite Line of Charge • From a table of integration:

22. Electric Potential Off the Axis of a Finite Line of Charge • Replacing this in the equation:

23. Electric Potential Off the Axis of a Finite Line of Charge • Should the point P lie along a line that is a perpendicular bisector of the line of charge as shown, you would integrate from the left end of the rod at x = -x to the right end of the rod at x = x.

24. Electric Potential Off the Axis of a Finite Line of Charge

25. Electric Potential Off the Axis of a Finite Line of Charge

26. Electric Potential of a Uniform Ring of Charge

27. Electric Potential of a Uniform Ring of Charge • Consider the ring as a line of charge that has been formed into a ring. • Divide the ring into equal elements of length dx containing an element of charge dq; each element of charge dq is the same distance r from point P. • Each element of charge dq can be considered as a point charge which contributes to the net electric potential at point P. • Each element of charge dq is the same distance r from point P.

28. Electric Potential of a Uniform Ring of Charge • Because electric potential is a scalar quantity, we do not need to consider the vector components from the elements of charge dq on the opposite side of the ring (180° away) as is required when determining the electric field. • For each element of charge dq:

29. Electric Potential of a Uniform Ring of Charge • The total electric potential V can be found by adding the contribution to the electric potential by each element of charge dq. • Integrate around the circumference of the ring: • K, a, and x are constants and are pulled out in front of the integral sign.

30. Electric Potential of a Uniform Ring of Charge • Left side of the integral: adding up all the contributions to the electric potential from each element of charge dq gives us the total electric potential.

31. Electric Potential of a Uniform Ring of Charge • Right side of the integral: • The electric potential at point P is: