Electric Potential with Integration

Potential Difference in a Variable E-field If E varies, we integrate the potential difference for a small displacement dl over the path from a to b ©2008 by W.H. Freeman and Company

Potential Difference in a Variable E-field Like gravity, the electric field is conservative. This means that the potential does not depend on the path taken from a to b. ©2008 by W.H. Freeman and Company

Potential vs. Potential Difference Like gravitational potential energy, only differences in electrical potential energy have physical meaning. We can choose a convenient reference level to be the zero of potential. The potential V of a point is the potential difference between that point and the reference level.

Example: Potential of a point charge Calculate the electric potential at a distance r away from a point charge q. We choose the reference level to be at infinity. We integrate along a radius, so dl = dr.

Quick Review: Potential The potential difference between two points is the change in potential energy per unit charge if a charge were moved from one point to the other. The potential difference can be found using the electric field. The potential of a point change is

Potential and Potential Energy We take the zero of potential energy the same as the zero of potential. Under those conditions, U E = qV, where q is the charged placed at the position with potential V. The formula chart gives the potential energy of charge q 2 located a distance r away from q 1 as

Superposition of Potentials If more than one charge is present, the potential at a point is the sum of the potentials of each charge. The formula on the formula chart is

Example- Potential of two charges In this arrangement, r 1 = |x| and r 2 = |x-a| ©2008 by W.H. Freeman and Company

Calculating V for Continuous Charge Distributions

Integrating over a charge distribution If instead of point charges, we have a distribution of charge, we treat each small element of charge as a point charge, and integrating over all the charge elements.

V on the Axis of a Ring ©2008 by W.H. Freeman and Company

Charge density Linear ( ) – = Q/L = dQ/dL Surface (  ) –  = Q/A = dQ/dA Volume (  ) –  = Q/V = dQ/dV

Electric Field Spherical symmetry E = 1 q 4  o r 2 Continuous charge distribution dE = 1 dq 4  o r 2

For linear distribution dE = 1 dq 4  o r 2 dE = 1 dx 4  o r 2 E =  E =  dE

For linear distribution P cylindrical symmetry in electric field

V near a plane of Charge We cannot do as we did for electric field, that is, calculate V for a disk, and then let the size of the disk grow to infinity. –This would yield an infinite potential. We can’t put the zero of potential at infinity, for the same reason. ©2008 by W.H. Freeman and Company

V near a plane of Charge Instead, we start with the electric field of a plane of charge, and integrate along a path from the plane to a distance x away. x We say the potential at the surface of the sheet is V 0. ©2008 by W.H. Freeman and Company

Potential near a plane of charge x ©2008 by W.H. Freeman and Company

Potential near a plane of charge ©2008 by W.H. Freeman and Company

Potential near a charged shell We consider first the potential at a point outside the shell. (r>R) ©2008 by W.H. Freeman and Company

Potential near a charged shell: Outside ©2008 by W.H. Freeman and Company

Potential near a charged shell: Outside Outside a charged shell, the potential is the same as for a point charge. ©2008 by W.H. Freeman and Company

Potential near a charged shell: Inside “Inside” the shell, there is no charge. The field is zero inside. If the field is zero, the potential cannot change, so V is what it is at the surface. ©2008 by W.H. Freeman and Company

Energy Storage

Potential Energy of a System of Charges Consider a system of two equal charges, q 1 and q 2. Putting the first charge in place requires no energy. Putting the second charge requires q 2 V, where V is the potential of the charges. q 2 is ½ the total charge Q, so the energy can also be written

Potential Energy of a System of Charges The potential energy of a system of charges q i is given by

Computing the Electric Field from the Potential The electric field points in the direction of greatest change in potential. In the one dimensional case,

Computing the Electric Field from the Potential In general,

Computing the Electric Field from the Potential In vector calculus notation