Chapter 7: Q13P (page 316)

A square loop of wire, with sides of length a , lies in the first quadrant of the xy plane, with one comer at the origin. In this region, there is a nonuniform time-dependent magnetic field $B (y, t) = {ky}^{3} t^{2} \hat{z}$ (where k is a constant). Find the emf induced in the loop.

Short Answer

Expert verified

The induced emf into loop is $- \frac{k t}{2} a^{5} .$

Step by step solution

Write the given data from the question.

The side of the square loop is a.

The square loop exists in xy plane.

The non-uniform time dependent magnetic field, $B (y, t) = {ky}^{3} t^{2} \hat{z}$

Calculate the induced emf in the loop.

Let’s take small strip dy on the square loop which at distance y from the x-axis.

The magnetic flux is given by,

$ϕ = B (y, t) . d A$

Here dA is the area of the strip.

Substitute $K y^{3} t^{2}$ for $B (y, t)$ and ady for dA into above equation.

$ϕ = k y^{3} t^{2} . a d y$

Integrate the above equation to calculate the flus through the entire loop.

$ϕ = \int_{0}^{a} k y^{3} t^{2} . a d y ϕ = k a t^{2} \int_{0}^{a} y^{3} d y ϕ = k a t^{2} {(\frac{y^{4}}{4})}_{0}^{a}$

Apply the limits,

$ϕ = k a t^{2} (\frac{a^{4}}{4} - \frac{0^{4}}{4}) ϕ = k a t^{2} (\frac{a^{4}}{4}) ϕ = \frac{k t^{2}}{4} a^{5}$

The induced emf in the loop is given by,

$e = - \frac{d ϕ}{d t}$

Substitute $\frac{k t^{2}}{4} a^{5}$ for $ϕ$ into above equation.

$e = - \frac{d}{d t} (\frac{k t^{2}}{4} a^{5}) e = - \frac{2 k t}{4} a^{5} e = - \frac{k t}{2} a^{5}$

Hence the induced emf into loop is $- \frac{k t}{2} a^{5}$ .

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A square loop of wire, with sides of length a , lies in the first quadrant of the xy plane, with one comer at the origin. In this region, there is a nonuniform time-dependent magnetic field $B (y, t) = {ky}^{3} t^{2} \hat{z}$ (where k is a constant). Find the emf induced in the loop.

Short Answer

Step by step solution

Write the given data from the question.

Calculate the induced emf in the loop.

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A square loop of wire, with sides of length a , lies in the first quadrant of the xy plane, with one comer at the origin. In this region, there is a nonuniform time-dependent magnetic field B(y,t)=ky3t2z^ (where k is a constant). Find the emf induced in the loop.

Short Answer

Step by step solution

Write the given data from the question.

Calculate the induced emf in the loop.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Thermodynamics

Electric Charge Field and Potential

Electricity and Magnetism

Measurements

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Astrophysics

Study anywhere. Anytime. Across all devices.

A square loop of wire, with sides of length a , lies in the first quadrant of the xy plane, with one comer at the origin. In this region, there is a nonuniform time-dependent magnetic field $B (y, t) = {ky}^{3} t^{2} \hat{z}$ (where k is a constant). Find the emf induced in the loop.