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Chapter 27: Problem 4

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

Expert verified
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

01

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.
02

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:
03

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.
04

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.
05

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.
06

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.
07

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.

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Most popular questions from this chapter

The magnitude of the magnetic force on a particle with charge \(-2 e\) moving with speed \(v=1.0 \cdot 10^{5} \mathrm{~m} / \mathrm{s}\) is \(3.0 \cdot 10^{-18} \mathrm{~N}\). What is the magnitude of the magnetic field component perpendicular to the direction of motion of the particle?

Show that the magnetic dipole moment of an electron orbiting in a hydrogen atom is proportional to its angular momentum, \(L: \mu=-e L / 2 m,\) where \(-e\) is the charge of the electron and \(m\) is its mass.

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It would be mathematically possible, for a region with zero current density, to define a scalar magnetic potential analogous to the electrostatic potential: \(V_{B}(\vec{r})=-\int_{\vec{r}_{0}}^{\vec{r}} \vec{B} \cdot d \vec{s},\) or \(\vec{B}(\vec{r})=-\nabla V_{B}(\vec{r}) .\) However, this has not been done. Explain why not.
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