PHYS 208 Lecture 6

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Conductors

Electric fields inside conductors is always 0; charge always resides on the surface of the conductor

Electric Potential

Review of Mechanics

Work of a force: $\int _{a}^{b}{\vec {F}}\cdot \mathrm {d} {\vec {l}}$

For conservative forces, $W=-(U_{b}-U_{a})=-\Delta U$

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Electric forces are conservative forces!

Application of Work to Coulomb's law:

$W=\int _{a}^{b}{\frac {kQq}{r^{2}}}\,{\hat {r}}\cdot \mathrm {d} {\vec {r}}=-\left({\frac {kQq}{r_{b}}}-{\frac {kQq}{r_{a}}}\right)$

Therefore, the potential energy of an electric charge is

$U(r)={\frac {kQq}{r}}+C$

Note: potential of multiple charges is the sum of potential energies between all possible pair combinations

$U=\sum _{ij}k\,{\frac {q_{i}q_{j}}{r_{ij}}}$

Because electricity is much stronger than gravity, the constant of integration $C$ actually matters now.

Conservation of energy also applies:

$W=\int ({\vec {F}}_{E}+{\vec {F}}_{\mathrm {agent} })\cdot \mathrm {d} {\vec {l}}=0$

Example

Moving a -1nc charge that is 1m away from a 5nc charge to a point 5m away:

$W=-[U(5m)-U(1m)]=-kq_{1}q_{2}\left[{\frac {1}{5m}}-{\frac {1}{1m}}\right]=-36\times 10^{-9}\mathrm {J}$

Work done my moving agent must be equal and opposite: 36 × 10⁻⁹ J (positive work — when the force and direction of movement are in the same direction)

Electric Potential for a point charge

Measured in J/C = volts [V]

$V(r)={\frac {kq}{r}}+C$

Based on the units, dividing work by the charge results in the integral of the electric field:

$V={\frac {W}{q}}=\int _{a}^{b}{\frac {\vec {F}}{q}}\cdot \mathrm {d} {\vec {l}}=\int _{a}^{b}{\vec {E}}\cdot \mathrm {d} {l}$

It is also important to note that the work divided by the charge is the negation of the change in electric potential:

$-\Delta V={\frac {W}{q}}\quad \longrightarrow \quad \Delta V=-\int _{a}^{b}{\vec {E}}\cdot \mathrm {d} {\vec {l}}$

In order to get rid of the constant, we can use infinity as a reference point where $V=0$:

$V(r)=V(r)-V(\infty )={\frac {kq}{r}}$

For a continuum (surface or solid) of point charges:

$V(r)=\int {\frac {kdq}{|{\vec {r}}-{\vec {r}}\,'|}}$