Chapter 29: Q. 51 (page 833)

The earth’s magnetic field, with a magnetic dipole moment of $8.0 \times 10^{22} A m^{2}$ , is generated by currents within the molten iron of the earth’s outer core. Suppose we model the core current as a $3000 k m$ -diameter current loop made from a 1000-km-diameter “wire.” The loop diameter is measured from the centers of this very fat wire.
a. What is the current in the current loop?
b. What is the current density $J$ in the current loop?
c. To decide whether this is a large or a small current density, compare it to the current density of a $1.0 A$ current in a $1.0 m m$ -diameter wire.

Short Answer

Expert verified

a. The current in the current loop is $1.13 \times 10^{10} A$ .

b. The current density $J$ in the current loop is $0.014 A / m^{2}$ .

c. $1.3 \times 10^{6} A / m^{2}$ The small wire has a very large current density compared to the large one.

Step by step solution

Part (a) Step 1 : Given Information

We need to find the current in the current loop.

Part (a) Step 2 : Simplify

The magnetic field dipole moment $M = 8 \times 10^{22} A m^{2}$ current-carrying loop is just like a magnetic dipole whose magnetic dipole :

$M = i \times A i = \frac{M}{A} i = \frac{8 \times 10^{22}}{π {(1.5 \times 10^{6})}^{2}} i = 1.13 \times 10^{10} A$

Part (b) Step 1 : Given Information.

We need to find density $J$ in the current loop.

Part (b) Step 2: Simplify

The current density $J = ?$

So,

role="math" localid="1650965922291" $i = J \times s J = \frac{i}{s} J = \frac{1.13 \times 10^{10}}{π {(0.5 \times 10^{6})}^{2}} J = 0.014 A / m^{2}$

Part (c) Step 1 : Given Information :

We need to find the current density of a $1.0 A$ current in a $1.0 m m$ -diameter wire.

Part (c) Step 2: Simplify

The wire with a diameter $1 m m$ , the radius is $r_{2} = 0.5 m m$ and the current flows is $I_{2} = 1 A$ .

We finding the $J_{2}$ as:

$J_{2} = \frac{I_{2}}{π r^{2}} = \frac{(1 A)}{π {(0.5 \times 10^{- 3} m)}^{2}} = 1.3 \times 10^{6} A / m^{2}$

The small wire has a very large current density comparing to the large one.

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Short Answer

Step by step solution

Part (a) Step 1 : Given Information

Part (a) Step 2 : Simplify

Part (b) Step 1 : Given Information.

Part (b) Step 2: Simplify

Part (c) Step 1 : Given Information :

Part (c) Step 2: Simplify

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