Solution 5: Fields Inside a Rotating Charged Cylinder
1. Magnetic Field ($B$):
A cylinder with surface charge density $\sigma$ rotating at angular velocity $\omega$ behaves like a solenoid with surface current density $K = \sigma v = \sigma (\omega R)$.
Given $\omega = kt$, the current density increases with time: $K = \sigma R k t$.
The magnetic field inside a long solenoid is uniform:
$$ B = \mu_0 K = \mu_0 \sigma R k t $$Observation: The magnetic field $B$ is uniform in space (inside the cylinder) but varies linearly with time $t$.
2. Electric Field ($E$):
A time-varying magnetic field induces an electric field according to Faraday’s Law: $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi}{dt}$.
Consider a circular loop of radius $r < R$ inside the cylinder:
$$ E (2\pi r) = \text{Area} \times \left| \frac{dB}{dt} \right| = (\pi r^2) (\mu_0 \sigma R k) $$ $$ E = \frac{1}{2} \mu_0 \sigma R k r $$Observation: The induced electric field $E$ is proportional to $r$ (non-uniform in space) but is independent of time (constant).
3. Energy Density ($u$):
The total energy density $u$ is the sum of magnetic and electric energy densities:
$$ u = u_B + u_E = \frac{B^2}{2\mu_0} + \frac{1}{2}\epsilon_0 E^2 $$Substituting the dependencies:
- $u_B \propto B^2 \propto t^2$
- $u_E \propto E^2 \propto (\text{constant})^2 = \text{constant}$ (for a fixed $r$)
Thus, $u(t) = a + b t^2$, where $a$ and $b$ are positive constants.
(b) Magnetic field is uniform but not constant; Electric field is constant but not uniform.
(d) Total energy density $u$ varies as $u = a + b t^2$.