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Chapter 13: Problem 1968

A small bar magnet has a magnetic moment $1.2 \mathrm{~A} \cdot \mathrm{m}^{2}\(. The magnetic field at a distance \)0.1 \mathrm{~m}$ on its axis will be tesla. (a) \(1.2 \times 10^{-4}\) (b) \(2.4 \times 10^{-4}\) (c) \(2.4 \times 10^{4}\) (d) \(1.2 \times 10^{4}\)

Short Answer

Expert verified
The magnetic field at a distance of \(0.1\,\mathrm{m}\) along the axis of the given small bar magnet is \(2.4 \times 10^{-4} \mathrm{T}\). Answer: (b) \(2.4 \times 10^{-4}\).

Step by step solution

01

Recall the formula for magnetic field due to a magnetic dipole

The magnetic field B due to a magnetic dipole at a point on its axis is given by the formula: \[B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{r^3}\] Where \(B\) is the magnetic field, \(\mu_0\) is the permeability of free space (equal to \(4\pi \times 10^{-7} \mathrm{T} \cdot \mathrm{m} / \mathrm{A}\)), \(M\) is the magnetic moment, and \(r\) is the distance from the dipole along its axis.
02

Substitute the given values into the formula

Given values are \(M = 1.2 \,\mathrm{A} \cdot \mathrm{m}^2\) and \(r = 0.1\,\mathrm{m}\). Substitute these values into the formula for the magnetic field: \[B =\frac{\mu_0}{4\pi} \cdot \frac{2(1.2\,\mathrm{A} \cdot \mathrm{m}^2)}{(0.1\,\mathrm{m})^3}\]
03

Calculate the magnetic field

Using \(\mu_0 = 4\pi \times 10^{-7} \mathrm{T} \cdot \mathrm{m} / \mathrm{A}\), let's plug in the value and calculate the magnetic field: \[B = \frac{(4\pi \times 10^{-7} \mathrm{T} \cdot \mathrm{m} / \mathrm{A})}{4\pi} \cdot \frac{2(1.2\,\mathrm{A} \cdot \mathrm{m}^2)}{(0.1\,\mathrm{m})^3}\] Simplify the expression: \[B = 10^{-7} \mathrm{T} \cdot \mathrm{m} / \mathrm{A} \cdot \frac{2.4\,\mathrm{A} \cdot \mathrm{m}^2}{0.001\,\mathrm{m}^3}\] \[B = 2.4 \times 10^{-4} \mathrm{T}\]
04

Match the calculated value with the given options

We find that the magnetic field at a distance of \(0.1\,\mathrm{m}\) along the axis of the given small bar magnet is \(2.4 \times 10^{-4} \mathrm{T}\), which corresponds to option (b). Answer: (b) \(2.4 \times 10^{-4}\)

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