Question

In the Hall effect, a potential difference produced across a conductor of finite thickness in a magnetic field by a current flowing through the conductor is given by a) the product of the density of electrons, the charge of an electron, and the conductor's thickness divided by the product of the magnitudes of the current and the magnetic field. b) the reciprocal of the expression described in part (a). c) the product of the charge on an electron and the conductor's thickness divided by the product of the density of electrons and the magnitudes of the current and the magnetic field. d) the reciprocal of the expression described in (c). e) none of the above.

Short answer

a. ne(d/IxB) b. 1/ne(d/IxB) c. e(d/nIxBy) d. 1/e(d/nIxBy) e. None of the above Answer: e. None of the above

Step-by-step solution

Define the Hall effect formula

$$V_H = \frac{I_x B_z d}{n e A}$$ Where \(V_H\) is the Hall voltage, \(I_x\) is the current flowing through the conductor in the x-direction, \(B_z\) is the magnetic field in the z-direction, \(d\) is the conductor's thickness, \(n\) is the density of electrons, \(e\) is the charge of an electron, and \(A\) is the cross-sectional area of the conductor.

Rewrite the formula

To better compare the given options with the Hall effect formula, rewrite the original formula as follows: $$V_H = \frac{d}{\frac{n e A}{I_x B_z}}$$ Now, let's analyze each option:

Option a

The potential difference is given by the product of: - density of electrons (\(n\)) - charge of an electron (\(e\)) - conductor's thickness (\(d\)) Divided by the product of: - magnitudes of the current (\(I_x\)) - the magnetic field (\(B_z\)) Comparing with the rewritten formula, this option is incorrect.

Option b

The potential difference is given by the reciprocal of the expression in part (a). Given that option a is not the correct formula, option b cannot be correct either.

Option c

The potential difference is given by the product of: - charge on an electron (\(e\)) - conductor's thickness (\(d\)) Divided by the product of: - density of electrons (\(n\)) - magnitudes of the current (\(I_x\)) - the magnetic field (\(B_z\)) Comparing with the rewritten formula, this option is incorrect.

Option d

The potential difference is given by the reciprocal of the expression in part (c). Given that option c is not the correct formula, option d cannot be correct either.

Option e

None of the above. As we saw that options a through d were not valid for the Hall effect formula, the right answer is option e, none of the above.