Chapter 16.2, Problem 52E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Experiments show that a steady current *I* in a long wire produces a magnetic field **B** that is tangent to any circle that lies in the plane perpendicular to the wire and whose center is the axis of the wire (as in the figure). *Ampère’s Law* relates the electric current to its magnetic effects and states that

∫_{C}**B** · *d***r*** = μ*_{0}*I*

where *I* is the net current that passes through any surface bounded by a closed curve *C*, and *μ*_{0} is a constant called the permeability of free space. By taking *C* to be a circle with radius *r*, show that the magnitude *B* = |**B**| of the magnetic field at a distance *r* from the center of the wire is

To determine

**To show:** The magnitude of the magnetic field at a distance *r* from the center of the wire as

Explanation

**Given data:**

Refer to the Figure at the bottom the question 52 in the textbook.

Write the equation of Ampereâ€™s law as follows.

Here,

**Formula used:**

Write the expression to find the unit tangent vector

Here,

Consider the parametric equations of the circle

Write the vector equation from the parametric equations as follows.

As the magnetic field

Here,

**Calculation of tangent vector**
**:**

To find the derivative of the vector function, differentiate each component of the vector function.

Differentiate each component of the vector function

Rewrite and compute the expression as follows.

**Calculation of unit tangent vector**
**:**

Substitute

Simplify the expression as follows.

Substitute

**Calculation of**
**:**

Differentiate the vector

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