A wire bent as a parabola y = kx ^ 2 is located in a uniform...

A wire bent as a parabola

y = {kx}^{2}

is located in a uniform magnetic field of induction

B

, the vector

B

being perpendicular to the plane

xy

. At

t = 0

, the sliding wire starts sliding from the vertex

O

with a constant acceleration a linearly as shown in Fig. Find the emf induced in the loop -

A wire bent as a parabola y = kx ^ 2 is located in a uniform magnetic field of induction B , the vector B being

A wire bent as a parabola y = kx ^ 2 is located in a uniform magnetic field of induction B , the vector B being

Options

A $By\sqrt{\dfrac{2a}{k}}$
B $By\sqrt{\dfrac{4a}{k}}$
C $By\sqrt{\dfrac{8a}{k}}$
D $By\sqrt{\dfrac{a}{k}}$

Correct answer

C. $By\sqrt{\dfrac{8a}{k}}$

Step-by-step solution

dϕ = B . dA = 2Bxdy and y = {kx}^{2}

∴ x = \sqrt{\dfrac{y}{k}}

∴ Emf E = \dfrac{dϕ}{dt} = 2B\sqrt{\dfrac{y}{k}}\dfrac{dy}{dt}

using

{v}^{2} = 2as

\dfrac{dy}{dt} = v = \sqrt{2ay}

or \left|ε\right| = \dfrac{dϕ}{dt} = 2B\sqrt{\dfrac{y}{k}}\sqrt{2ay}

<br/><br/>or   \left|ε\right| =By\sqrt{\dfrac{8a}{k}}

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