A conducting square frame of side and a long straight wire carrying current are located in the same plane as shown in the figure. The frame moves to the right with a constant velocity . The emf induced in the frame will be proportional to [201

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Q.
A conducting square frame of side $' a'$ and a long straight wire carrying current $I$ are located in the same plane as shown in the figure. The frame moves to the right with a constant velocity $' V'$ . The emf induced in the frame will be proportional to [2015]

1 $\frac{1}{{(2 x + a)}^{2}}$

2 $\frac{1}{(2 x - a) (2 x + a)}$

3 $\frac{1}{x^{2}}$

4 $\frac{1}{{(2 x - a)}^{2}}$

Ans.
(2)

Here, $P Q = R S = P R = Q S = a$

Emf induced in the frame, $ε = B_{1} (PQ) V - B_{2} (RS) V$

$= \frac{μ_{0} I}{2 π (x - a / 2)} a V - \frac{μ_{0} I}{2 π (x + a / 2)} a V$

$= \frac{μ_{0} I}{2 π} [\frac{2}{(2 x - a)} - \frac{2}{(2 x + a)}] a V$

$= \frac{μ_{0} I}{2 π} \times 2 [\frac{2 a}{(2 x - a) (2 x + a)}] a V$

$∴$ $ε \propto \frac{1}{(2 x - a) (2 x + a)}$

Solution

Q.
A conducting square frame of side $' a'$ and a long straight wire carrying current $I$ are located in the same plane as shown in the figure. The frame moves to the right with a constant velocity $' V'$ . The emf induced in the frame will be proportional to [2015]

1 $\frac{1}{{(2 x + a)}^{2}}$

2 $\frac{1}{(2 x - a) (2 x + a)}$

3 $\frac{1}{x^{2}}$

4 $\frac{1}{{(2 x - a)}^{2}}$

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Solution

Q. A conducting square frame of side 'a' and a long straight wire carrying current I are located in the same plane as shown in the figure. The frame moves to the right with a constant velocity 'V'. The emf induced in the frame will be proportional to [2015]

1 1(2x+a)2

2 1(2x-a)(2x+a)

3 1x2

4 1(2x-a)2

Ans. (2) Here, PQ=RS=PR=QS=a Emf induced in the frame, ε=B1(PQ)V-B2(RS)V =μ0I2π(x-a/2)aV-μ0I2π(x+a/2)aV =μ0I2π[2(2x-a)-2(2x+a)]aV =μ0I2π×2[2a(2x-a)(2x+a)]aV ∴ ε∝1(2x-a)(2x+a)

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Q.
A conducting square frame of side $' a'$ and a long straight wire carrying current $I$ are located in the same plane as shown in the figure. The frame moves to the right with a constant velocity $' V'$ . The emf induced in the frame will be proportional to [2015]

1 $\frac{1}{{(2 x + a)}^{2}}$

2 $\frac{1}{(2 x - a) (2 x + a)}$

3 $\frac{1}{x^{2}}$

4 $\frac{1}{{(2 x - a)}^{2}}$