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Chapter 29: Q. 23 (page 831)

The value of the line integral of $\overset{鈫�}{B}$ around the closed path in FIGURE EX29.23 is $1.38 脳 10^{- 5} T . m$ . What are the direction (in or out of the page) and magnitude of $I_{3}$ ?

Short Answer

Expert verified

The current $I_{3} = 23 A$ and it is points into the paper.

Step by step solution

01

Given information

We have given,

line integral = $1.38 脳 10^{- 5} T . m$

We have to find the current $I_{3}$ and its flow of direction.

02

Simplify

Using ampere's law, for any closed path is given by,

$鈭� \overset{鈫�}{B} . \overset{鈫�}{d S} = 渭_{0} I$

Where I is a total enclosed current in the magnetic field curve.

then total current will be

$I = I_{2} + I_{3} I = - 12 A + I_{3} . . . . . . . . . . . . . . . . . . . . . . . (1)$

since other currents are out of closed curve.

since,

$鈭� \overset{鈫�}{B} . \overset{鈫�}{d S} = 渭_{0} I 鈭� \overset{鈫�}{B} . \overset{鈫�}{d S} = (4 蟺脳 10^{- 7} T . m / A) I I = \frac{鈭� \overset{鈫�}{B} . \overset{鈫�}{d S}}{4 蟺脳 10^{- 7}} = \frac{1.38 脳 10^{- 5} T . m}{4 脳 3.14 脳 10^{- 7} T . m / A} I = 11 A$

using equation (1),

localid="1649421657359" $I = 11 A = - 12 A + I_{3} I_{3} = 23 A .$

the positive sine indicates that current will be into the paper.

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Most popular questions from this chapter

A proton moves in the magnetic field $\overset{鈫�}{B} = 0.50 \hat{谋} T$ with a speed of $1.0 脳 10^{7} m / s$ in the directions shown in FIGURE. For each, what is magnetic force $\overset{鈫�}{F}$ on the proton? Give your answers in component form.

To five significant figures, what are the cyclotron frequencies in a $3.0000 T$ magnetic field of the ions $(a) {O_{2}}^{+}$ , $(b) {N_{2}}^{+}$ , and $(c) {CO}^{+}$ ? The atomic masses are shown in the table; the mass of the missing electron is less than $0.001$ u and is not relevant at this level of precision. Although ${N_{2}}^{+}$ and ${C_{2}}^{+}$ both have a nominal molecular mass of $28$ , they are easily distinguished by virtue of their slightly different cyclotron frequencies. Use the following constants: $1 u = 1.6605 脳 10^{- 27} K g, e = 1.6022 脳 10^{- 19} C$ .

a. An infinitely long sheet of charge of width L lies in the x y plane between x=-L / 2 and x=L / 2. The surface charge density is \eta. Derive an expression for the electric field $E$ at height $z$ above the centerline of the sheet.

b. Verify that your expression has the expected behavior if $z 鈮� L$ and if $z 鈮� L$

c. Draw a graph of field strength E versus z.

A long, straight conducting wire of radius R has a nonuniform current density $J = \frac{J_{0} r}{R}$ , where $J_{0}$ is a constant. The wire carries total current I.

a. Find an expression for $J_{0}$ in terms of I and R.

b. Find an expression for the magnetic field strength inside the wire at radius r.

c. At the boundary, r = R, does your solution match the known field outside a long, straight current-carrying wire?

The element niobium, which is a metal, is a superconductor (i.e., no electrical resistance) at temperatures below $9 K$ . However, the superconductivity is destroyed if the magnetic field at the surface of the metal reaches or exceeds $0.10 T$ . What is the maximum current in a straight, $3 mm$ diameter superconducting niobium wire?

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