EMI BYU 2

2. Surface Charge Density on Moving Bar

Part (a): Perfect Conductor

The magnetic force on charge carriers is $F_m = q(\vec{v} \times \vec{B})$. This drives charges to the surfaces until the accumulated surface charge creates an electric field $\vec{E}$ that exactly cancels the magnetic force inside the conductor ($F_{net} = 0$).

$$ qE = qvB \implies E = vB $$

Using Gauss’s Law at the surface, the electric field just inside the surface charge layer is $E = \sigma / \epsilon_0$.

$$ \frac{\sigma}{\epsilon_0} = vB \implies \sigma = \epsilon_0 vB $$

Direction: For standard $v$ and $B$, positive charges accumulate on the lower face.

Part (b): Dielectric Material

In a dielectric, there are no free charges. The magnetic force polarizes the material, creating bound surface charges $\sigma_p$ (Polarization). These charges create a depolarization field $E_p$ inside the material that opposes the motional field.

The net internal electric field is $E_{in} = vB – E_p$, where $E_p = \frac{\sigma_p}{\epsilon_0}$.

The polarization $P$ is given by $P = \sigma_p$. From dielectric theory:

$$ P = \epsilon_0 (\epsilon_r – 1) E_{in} $$ $$ \sigma_p = \epsilon_0 (\epsilon_r – 1) \left( vB – \frac{\sigma_p}{\epsilon_0} \right) $$ $$ \sigma_p = \epsilon_0 (\epsilon_r – 1) vB – (\epsilon_r – 1) \sigma_p $$ $$ \sigma_p [1 + (\epsilon_r – 1)] = \epsilon_0 (\epsilon_r – 1) vB $$ $$ \sigma_p \epsilon_r = \epsilon_0 (\epsilon_r – 1) vB $$