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Chapter 16: Problem 26

A flat coil carrying a current has a magnetic moment \(\vec{\mu}\). It is placed in a magnetic field \(\vec{B}\). The torque on the coil is \(\vec{\tau}\), then (A) \(\vec{\tau}=\vec{\mu} \cdot \vec{B}\) (B) \(\vec{\tau}=\vec{B} \times \vec{\mu}\) (C) \(|\vec{\tau}|=\vec{\mu} \cdot \vec{B}\) (D) \(\vec{\tau}\) is perpendicular to both \(\vec{\mu}\) and \(\vec{B}\).

Short Answer

Expert verified
The correct options are (B) \(\vec{\tau} = \vec{B} \times \vec{\mu}\) and (D) \(\vec{\tau}\) is perpendicular to both \(\vec{\mu}\) and \(\vec{B}\), as \(\vec{\tau}\) is the cross-product of the magnetic moment \(\vec{\mu}\) and the magnetic field \(\vec{B}\).

Step by step solution

01

1. Formula for torque in a magnetic field

The formula for the torque \(\vec{\tau}\) on a current loop with magnetic moment \(\vec{\mu}\) in a magnetic field \(\vec{B}\) is given by: \[\vec{\tau} = \vec{\mu} \times \vec{B}\]
02

2. Compare the formula with given options

Now, let's compare the formula with each given option: (A) \(\vec{\tau} = \vec{\mu} \cdot \vec{B}\) This option is incorrect, as the torque is the cross-product of the magnetic moment and the magnetic field, not the dot product. (B) \(\vec{\tau} = \vec{B} \times \vec{\mu}\) This option is correct, as the torque is indeed the cross-product of the magnetic moment and the magnetic field. (C) \(|\vec{\tau}| = \vec{\mu} \cdot \vec{B}\) This option is incorrect, as scalar magnitudes of the torque and the magnetic moment are not equal to their dot product. (D) \(\vec{\tau}\) is perpendicular to both \(\vec{\mu}\) and \(\vec{B}\) This option is correct. Since the torque is the cross product of the magnetic moment and the magnetic field, it is by definition perpendicular to both of these vectors. So the correct options are (B) and (D).

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Most popular questions from this chapter

When the current changes from \(+2 \mathrm{~A}\) to \(-2 \mathrm{~A}\) in \(0.05 \mathrm{~s}\), an EMF of \(8 \mathrm{~V}\) is induced in a coil. The coefficient of self-induction of the coil is (A) \(0.1 \mathrm{H}\) (B) \(0.2 \mathrm{H}\) (C) \(0.4 \mathrm{H}\) (D) \(0.8 \mathrm{H}\)

Two coils are placed close to each other. The mutual inductance of the pair of coils depends upon (A) the rates at which currents are changing in the two coils. (B) relative position and orientation of the two coils. (C) the materials of the wires of the coils. (D) the currents in the two coils.

A capacitor of capacitance \(C=\frac{18}{\pi} \mathrm{mH}\) having initial charge \(Q_{0}\) connected to an inductor of inductance \(L=\frac{18}{\pi} \mathrm{mH}\) at \(t=0\). Find the time (in milli second) after energy stored in electric field is three times energy stored in magnetic field.

A conducting rod \(A B\) moves parallel to \(x\)-axis in the \(x-y\) plane. A uniform magnetic field \(B\) pointing normally out of the plane exists throughout the region. A force \(F\) acts perpendicular to the rod, so that the rod moves with uniform velocity \(v\). The force \(F\) is given by (neglect resistance of all the wires). (A) \(\frac{v B^{2} l^{2}}{R} e^{-t / R C}\) (B) \(\frac{v B^{2} l^{2}}{R}\) (C) \(\frac{v B^{2} l^{2}}{R}\left(1-e^{-t / R C}\right)\) (D) \(\frac{v B^{2} l^{2}}{R}\left(1-e^{-2 t / R C}\right)\)

A uniform conducting ring of mass \(\pi \mathrm{kg}\) and radius \(1 \mathrm{~m}\) is kept on smooth horizontal table. A uniform but time varying magnetic field \(B=\left(\hat{i}+t^{2} \hat{j}\right)\) Tesla is present in the region. (Where \(t\) is time in seconds). Resistance of ring is \(2 \Omega .\) Then, \(\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)\) Time (in second) at which ring start toppling is (A) \(\frac{10}{\pi}\) (B) \(\frac{20}{\pi}\) (C) \(\frac{5}{\pi}\) (D) \(\frac{25}{\pi}\)
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