Chapter 30: Q75P (page 901)

A rectangular loop of N closely packed turns is positioned near a long straight wire as shown in Figure. What is the mutual inductance M for the loop–wire combination if $N = 100, a = 1.0 c m, b = 8.0 c m, a n d l = 30 c m$

Short Answer

Expert verified

Mutual Inductance M for the loop – wire combination will be $1.32 \times 10^{- 5} H$

Step by step solution

Given

$N = 100 a = 1.0 cm = 0.01 m b = 8.0 cm = 0.08 m l = 30 cm = 0.3 m$

Understanding the concept

We know the formula for the magnetic flux in terms of the magnetic field. We also have the formula for the mutual inductance in terms of the magnetic flux. So, we can calculate the required answer using the given information.

Formula:

$ϕ = \int B d A B = \frac{μ_{0} I}{2 πr} M = \frac{N ϕ}{I}$

Calculate mutual Inductance M for the loop – wire combination

We have,

$ϕ = \int B d A$

Magnetic flux through the loop can be written as,

$ϕ_{B} = \int_{a}^{a + b} B_{w i r e} \times l d r ϕ_{B} = \int_{a}^{a + b} (\frac{μ_{0} I}{2 πr}) \times l d r ϕ_{B} = \frac{μ_{0} I l}{2 π} \int_{a}^{a + b} (\frac{1}{r}) d r ϕ_{B} = \frac{μ_{0} I l}{2 π} {[I n r]}_{a}^{a + b} ϕ_{B} = \frac{μ_{0} I l}{2 π} (l n (a + b) - l n a) ϕ_{B} = \frac{μ_{0} I l}{2 π} \times (I n (1 + \frac{b}{a}))$

We know that,

$M = \frac{N ϕ}{I} M = \frac{N}{I} \times (\frac{μ_{0} I l}{2 π} \times (I n (1 + \frac{b}{a}))) M = \frac{N μ_{0} I}{2 π} \times (I n (1 + \frac{0.08}{0.01})) M = 1.32 \times 10^{- 5} H$

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A rectangular loop of N closely packed turns is positioned near a long straight wire as shown in Figure. What is the mutual inductance M for the loop–wire combination if $N = 100, a = 1.0 c m, b = 8.0 c m, a n d l = 30 c m$

Short Answer

Step by step solution

Given

Understanding the concept

Calculate mutual Inductance M for the loop – wire combination

One App. One Place for Learning.

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A rectangular loop of N closely packed turns is positioned near a long straight wire as shown in Figure. What is the mutual inductance M for the loop–wire combination if N=100,a=1.0cm,b=8.0cm,andl=30cm

Short Answer

Step by step solution

Given

Understanding the concept

Calculate mutual Inductance M for the loop – wire combination

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fundamentals of Physics

Oscillations

Electric Charge Field and Potential

Atoms and Radioactivity

Rotational Dynamics

Dynamics

Study anywhere. Anytime. Across all devices.

A rectangular loop of N closely packed turns is positioned near a long straight wire as shown in Figure. What is the mutual inductance M for the loop–wire combination if $N = 100, a = 1.0 c m, b = 8.0 c m, a n d l = 30 c m$