Chapter 29: Q13P (page 857)

In Fig. 29-44 point $P_{1}$ is at distance $R = 13.1 cm$ on the perpendicular bisector of a straight wire of length $L = 18.0 cm$ . carrying current. (Note that the wire is notlong.) What is the magnitude of the magnetic field at $P_{1}$ due to i?

Short Answer

Expert verified

The magnitude of the magnetic field at $P_{1}$ due to i is $5.03 \times 10^{- 8} T$

Step by step solution

Given

Current $i = 58.2 mA$
Distance of point P on the perpendicular bisector of wire $R = 13.1 cm = 0.131 m$
Length of wire $L = 18.0 cm = 0.18 m$

Determine the formulas:

Use the formula of Biot-Savarts law to find the magnitude of the field at $P_{1}$ due to i

Formula:

$d B = \frac{μ_{0} i d l}{4 π} \frac{s i n θ}{r^{2}}$

Calculate the magnitude of the magnetic field at P1 due to i

According to Bio-Savarts law

$d B = \frac{μ_{0} i d l}{4 π} \frac{\sin θ}{r^{2}}$

If r makes an angle $θ$ withl then

$r = \sqrt{l^{2} + R^{2}}$

And

$\sin θ = \frac{R}{R} = \frac{R}{\sqrt{l^{2} + R^{2}}} l = \frac{L}{2} = \frac{0.18 m}{2} = 0.09 m$

Integrating an equation of Bio-Savarts law

$B = \int_{0}^{0.09} d B = \int_{0}^{0.09} \frac{μ_{0} i d l}{4 π} \frac{\sin θ}{r^{2}} B = \frac{μ_{0} i}{4 π} \int_{0}^{0.09} \frac{\sin θ d l}{r^{2}} B = \frac{μ_{0} i}{4 π} \int_{0}^{0.09} \frac{R}{\sqrt{l^{2} + R^{2}}} \frac{d l}{{(\sqrt{l^{2} + R^{2}})}^{2}} B = \frac{μ_{0} i R}{4 π} \int_{0}^{0.09} \frac{d l}{{(l^{2} + R^{2})}^{3 / 2}}$

Solve further as:

$B = \frac{μ_{0} i R}{4 π} \frac{1}{R^{2}} {|\frac{l}{{(l^{2} + R^{2})}^{1 / 2}}|}_{0}^{0.09} B = \frac{μ_{0} i}{2 π R} [\frac{l}{{(l^{2} + R^{2})}^{1 / 2}}] B = \frac{(4 π \times 10^{- 7} \frac{Tm}{A}) (0.0582 A)}{2 π (0.131 m)} [\frac{0.09 m}{{[{(0.09 m)}^{2} + {(0.131 m)}^{2}]}^{1 / 2}}] B = 5.03 \times 10^{- 8} T$

Therefore, the magnitude of the magnetic field at $P_{1}$ due to i is $5.03 \times 10^{- 8} T$

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In Fig. 29-44 point $P_{1}$ is at distance $R = 13.1 cm$ on the perpendicular bisector of a straight wire of length $L = 18.0 cm$ . carrying current. (Note that the wire is notlong.) What is the magnitude of the magnetic field at $P_{1}$ due to i?

Short Answer

Step by step solution

Given

Determine the formulas:

Calculate the magnitude of the magnetic field at P1 due to i

One App. One Place for Learning.

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In Fig. 29-44 point P1is at distance R=13.1cmon the perpendicular bisector of a straight wire of length L=18.0cm. carrying current. (Note that the wire is notlong.) What is the magnitude of the magnetic field at P1due to i?

Short Answer

Step by step solution

Given

Determine the formulas:

Calculate the magnitude of the magnetic field at P1 due to i

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electromagnetism

Fields in Physics

Work Energy and Power

Kinematics Physics

Torque and Rotational Motion

Translational Dynamics

Study anywhere. Anytime. Across all devices.

In Fig. 29-44 point $P_{1}$ is at distance $R = 13.1 cm$ on the perpendicular bisector of a straight wire of length $L = 18.0 cm$ . carrying current. (Note that the wire is notlong.) What is the magnitude of the magnetic field at $P_{1}$ due to i?