Question

A strip of copper \(2.0 \mathrm{cm}\) wide carries a current \(I=\) $30.0 \mathrm{A}\( to the right. The strip is in a magnetic field \)B=5.0 \mathrm{T}$ into the page. (a) What is the direction of the average magnetic force on the conduction electrons? (b) The Hall voltage is \(20.0 \mu \mathrm{V}\). What is the drift velocity?

Short answer

The direction of the average magnetic force on the conduction electrons is upwards. (b) What is the drift velocity of the conduction electrons in the copper strip considering the Hall voltage? The drift velocity of the conduction electrons in the copper strip is \(1.110\times 10^{-2}\) cm/s.

Step-by-step solution

Determine the direction of the magnetic force on the conduction electrons

To determine the direction of the magnetic force, we use the right-hand rule. First, point your right thumb in the direction of the current (to the right), then point your fingers in the direction of the magnetic field (into the page). The palm of your hand will now face in the direction of the magnetic force on the conduction electrons.

Apply the right-hand rule

Following the right-hand rule, when pointing your thumb to the right and fingers into the page, you will find that the palm of your hand points upwards. Therefore, the direction of the average magnetic force on the conduction electrons is upwards.

Calculate the drift velocity using the Hall voltage

To find the drift velocity, we need to use the given Hall voltage. The Hall voltage V_H is related to the magnetic field, current, and strip width by \(V_H = \frac{IB}{ne}\) where \(I\) is the current, \(B\) is the magnetic field, \(n\) is the charge density of electrons, and \(e\) is the elementary charge. First, we need to find the charge density n for copper. The charge density can be calculated by multiplying the number of free electrons per atom (z) and the atomic mass density (# of atoms per {[}cm^3{]}). For copper: z = 1 (assumption: one free electron per atom), Atomic mass = \(m = 63.5\text{ g/mol}\), Density = \(ρ = 8.92\text{ g/cm}^3\), and Avogadro's number: \(N_A = 6.022 \times 10 ^ {23}\text{ mol}^{-1}\) \(n = \frac{zρN_A}{m}\)

Calculate the charge density for copper

Substituting the values in the formula, we get: \(n = \frac{(1)(8.92)(6.022\times 10^{23})}{63.5}\) \(n = 8.476 {\times} 10^{22}\text{/cm}^3\)

Calculate the drift velocity

Finally, we can rearrange the Hall voltage formula to obtain the drift velocity, v: \(v = \frac{IB}{neV_H}\) Substitute the values to get, \(v = \frac{(30)(5)}{(8.476\times 10^{22})(1.6\times 10^{-19})(20\times 10^{-6})}\) \(v = 1.110\times 10^{-2}\text{ cm/s}\) The drift velocity of the conduction electrons in the copper strip is \(1.110\times 10^{-2}\) cm/s.