Chapter 21 Electric Potential

Conservation of energy Conservative force Conservation of energy + charged Potential (energy) air → conductor Potential difference (voltage) discharge 2

Coulomb force is conservative Comparing : Electrostatic / Coulomb force is conservative Work doesn’t depend on the path: The circulation theorem of electric field 3

Define electric potential: Conservative force → potential energy U U depends on the charge! Define electric potential: + - Potential difference / voltage: 4

Electric potential & electric field Difference in potential energy between a and b: a b If position b is the zero point: Relationship between electric field and potential Unit of electric potential: Volt (V) 5

It depends on the chosen of zero point Electric potential: It depends on the chosen of zero point Usually let V = 0 at infinity: For the field created by a point charge : Q . a 6

Properties of electric potential 1) Va → potential energy per unit (+) charge 2) Va depends on the zero point, Vba does not 3) Va and Vba are scalars, differ from E vector 7

Calculation of electric potential If the electric field is known: Potential of point charge: (V = 0 at infinity) System of several point charges: r . A 8

Potential of electric dipole Example1: (a) Determine the potential at o, a, b. (b) To move charge q from a to b, how much work must be done? Solution: (a) . h l (b) 9

. Charged conductor sphere Example2: Determine the potential at a distance r from the center of a uniformly charged conductor sphere (Q, R) for (a) r > R; (b) r = R; (c) r < R. . a + Solution: All charges on surface: Q (a) V at r > R: Same as point charge 10

Same potential at any point! . b (b) V at r = R: + Q . c (c) V at r < R: Same potential at any point! Discussion: + - ? (1) Conductor: equipotential body (2) Charges on holey conductor 11

Uniformly charged rod Example3: A thin rod is uniformly charged (λ, L). Determine the potential for points along the line outside of the rod. Solution: Choose infinitesimal charge dQ Potential at point b? . b dx dQ x a d L 12

Charged ring Example4: A thin ring is uniformly charged (Q, R). Determine the potential on the axis. Solution: R x Q . a r dQ or: Discussion: (1) Not uniform? (2) x >> R (3) other shapes 13

Conical surface Question: Charges are uniformly distributed on a conical surface (σ, h, R). Determine the electric potential at top point a. (V = 0 at infinity) h R a 14

Electrostatic induction Question: Put a charge q nearby a conductor sphere initially carrying no charge. Determine the electric potential on the sphere. - + Q R b q o Connect to the ground? 15

Air can become ionized due to high electric field *Breakdown voltage Air can become ionized due to high electric field Air → conducting → charge flows Breakdown of air: Breakdown voltage for a spherical conductor? Q R 16

Equipotential surfaces Visualize electric potential: equipotential surfaces Points on surface → same potential (1) Surfaces ⊥ field at any point. (2) Move on surface, no work done (3) Move along any field line, V ↘ (4) Surfaces dense → strong field 17

(1) Equipotential surfaces for dipole system: Some examples (1) Equipotential surfaces for dipole system: (2) Conductors (3) Parallel but not uniform field lines? 18

To determine electric field from the potential, use E determined from V To determine electric field from the potential, use In differential form: Component of E in the direction of dl : Total electric field: 19

Gradient → partial derivative in direction V changes most rapidly Gradient of V V V+dV 1 2 dl dn  a b Gradient of V : Gradient → partial derivative in direction V changes most rapidly 20

Determine E from V Example5: The electric potential in a region of space varies as V = x2yz. Determine E. Solution: 21

Example6: The electric field in space varies as . Determine V. Determine V from E Example6: The electric field in space varies as . Determine V. Solution: ( C is constant ) 22