Question 97: Current Electricity

Question

An infinite line charge of uniform electric charge density l lies along the axis of an electrically conducting infinite cylindrical shell of radius R. At time t = 0, the space inside the cylinder is filled with a material of permittivity e and electrical conductivity s. The electrical conduction in the material follows Ohm's law. Which one of the following graphs best describes the subsequent variation of the magnitude of current density j (t) at any point in the material?

Accepted Answer

Let the charge per unit length on the axis be $$\lambda$$(t) at a time t. The electric field due to this line charge at a radial distance r is given by
$$\overrightarrow E (r,t) = {{\lambda (t)} \over {2\pi  \in r}}\widehat r$$. ...... (1)
The free charges inside the conductor start moving radially due to the presence of electric field. The current density at a point is defined as the current flowing across a unit area placed perpendicular to the direction of current flow. The current density at a point is related to the electric field at that point by Ohm's law i.e.,
$$\overrightarrow j (r,t) = \sigma \overrightarrow E (r,t)$$, ....... (2)
where $$\sigma$$ is the electrical conductivity of the conductor. The current through a cylindrical shell of radius r and length l is given by
$$I = \int_{surface} {\overrightarrow j (r,t)\,.\,d\overrightarrow A  = j(r,t)(2\pi rl)} $$. ...... (3)
By conservation of charge, the current I is equal to the rate of decrease of charge q on axial line segment of length l i.e.,
$$I =  - {{dq} \over {dt}} =  - l{{d\lambda (t)} \over {dt}}$$. ...... (4)
From equations (1) - (4)
$${{d\lambda (t)} \over {dt}} =  - {{\sigma \lambda (t)} \over  \in }$$.
Integrate with initial condition $$\lambda$$(t = 0) = $$\lambda$$
0
to get
$$\lambda (t) = {\lambda _0}\exp {e^{ - {{\sigma t} \over  \in }}}$$,
Substitute $$\lambda$$(t) in equation (2) to get
$$\overrightarrow j (r,t) = {{\sigma {\lambda _0}} \over {2\pi  \in r}}{e^{ - {{\sigma t} \over  \in }}}$$.
Note that current density varies as 1/r with radial distance r. It decreases exponentially with time and becomes zero at t $$	o$$ $$\infty$$.

Answer

Answer

Answer

Answer

Current Electricity

Choose the correct answer: