Question

(a) Define current density. Scalar or vector? E field in a conductor (n electrons/volume, mass m, charge -e, relaxation time τ). Show j = αE, where α = ne²/mτ. (b) What's a Wheatstone bridge? Balance conditions?

Hint:

(a) Drift velocity and current density. (b) Balanced bridge, no current.

Answer 1

(a) Current Density

Current density \( \mathbf{j} \) is the current per unit area and is a vector. It is given by: 

\[ \mathbf{j} = \frac{I}{A} \]

Derivation

The drift velocity \( v_d \) is related to the electric field \( E \) as:

\[ v_d = -\frac{eE\tau}{m} \]

Where \( e \) is the charge of an electron, \( \tau \) is the relaxation time, and \( m \) is the mass of the electron.

The current density is given by the product of the number of charge carriers \( n \), the charge of the carriers \( e \), and their drift velocity \( v_d \):

\[ \mathbf{j} = n(-e)v_d = n \cdot (-e) \cdot \left( -\frac{eE\tau}{m} \right) \]

Simplifying this, we get:

\[ \mathbf{j} = \frac{ne^2\tau E}{m} \]

Thus, the conductivity \( \alpha \) is:

\[ \alpha = \frac{ne^2\tau}{m} \]

(b) Wheatstone Bridge

The Wheatstone bridge is used to measure an unknown resistance. It is balanced when the following condition is met:

\[ \frac{R_1}{R_2} = \frac{R_3}{R_4} \]

In this case, there is no current flowing through the galvanometer, and the bridge is considered balanced.