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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^b \vec{F} \cdot \mathrm{d}\vec{l}}

For conservative forces, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W = -(U_b - U_a) = -\Delta U}

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Electric forces are conservative forces!

Application of Work to Coulomb's law:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(r) = \frac{kQq}{r} + C}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U = \sum_{ij} k \, \frac{q_iq_j}{r_{ij}}}

Because electricity is much stronger than gravity, the constant of integration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} actually matters now.

Conservation of energy also applies:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W = -[U(5m) - U(1m)] = -kq_1q_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]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r) = \frac{kq}{r} + C}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = 0} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r) = V(r) - V(\infty) = \frac{kq}{r}}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r) = \int \frac{kdq}{|\vec{r}-\vec{r}\,'|}}