Question 58: Electromagnetic Induction

Question

A 1 m long metal rod AB completes the circuit as shown in figure. The area of circuit is perpendicular to the magnetic field of 0.10 T . If the resistance of the total circuit is $2 \Omega$ then the force needed to move the rod towards right with constant speed $(v)$ of $1.5 \mathrm{~m} / \mathrm{s}$ is $\_\_\_\_$ N.

Accepted Answer

When a conductor moves through a uniform magnetic field, it experiences motional EMF. For a rod of length l moving with velocity v perpendicular to a magnetic field B , the induced emf is :
$$ \varepsilon=\mathrm{Bvl} $$
The given values are,
$\mathrm{B}=0.10 \mathrm{~T}$
$\mathrm{v}=1.5 \mathrm{~m} / \mathrm{s}$
$\mathrm{l}=1 \mathrm{~m}$
Putting the values,
$$ \varepsilon=(0.10) 	imes(1.5) 	imes(1)=0.15 	ext { Volts } $$
Using Ohm's Law, where $R$ is the total resistance of the circuit:
$$ I=\frac{\varepsilon}{R} $$
Given $\mathrm{R}=2 \Omega$ :
$$ I=\frac{0.15}{2}=0.075 	ext { Amperes } $$
A current-carrying wire in a magnetic field experiences a force given by:
$$ F_m=I l B \sin (	heta) $$
Since the rod is perpendicular to the field, $	heta=90^{\circ}$,
$$ \Rightarrow \mathrm{F}_{\mathrm{m}}=\mathrm{IlB} $$
According to Lenz's Law, this magnetic force will act in a direction opposite to the motion (towards the left).
$$ \mathrm{F}_{\mathrm{m}}=(0.075) 	imes(1) 	imes(0.10) $$
$$\Rightarrow $$ $$ \mathrm{F}_{\mathrm{m}}=0.0075 \mathrm{~N}=7.5 	imes 10^{-3} \mathrm{~N} $$
Then the net force to move the rod with constant speed must be zero. Therefore, the external force ( $\mathrm{F}_{	ext {ext }}$ ) must be equal in magnitude to the magnetic force:
$$ \mathrm{F}_{\mathrm{ext}}=\mathrm{F}_{\mathrm{m}}=7.5 	imes 10^{-3} \mathrm{~N} $$
Therefore, the external force required to maintain the constant velocity of $1.5 \mathrm{~m} / \mathrm{s}$ against the magnetic resistance is $7.5 	imes 10^{-3} \mathrm{~N}$.
Hence, the correct option is (B).

Answer

$5.7 	imes 10^{-2}$

Answer

$7.5 	imes 10^{-3}$

Answer

$5.7 	imes 10^{-3}$

Answer

$7.5 	imes 10^{-2}$

Electromagnetic Induction

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