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Chapter 29: Q. 51 (page 833)

The earth鈥檚 magnetic field, with a magnetic dipole moment of 8.0脳1022Am2, is generated by currents within the molten iron of the earth鈥檚 outer core. Suppose we model the core current as a 3000km-diameter current loop made from a 1000-km-diameter 鈥渨ire.鈥� 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 1.0A current in a 1.0mm-diameter wire.

Short Answer

Expert verified

a. The current in the current loop is 1.13脳1010A.

b. The current density Jin the current loop is 0.014A/m2.

c.1.3脳106A/m2The small wire has a very large current density compared to the large one.

Step by step solution

01

Part (a) Step 1 : Given Information

We need to find the current in the current loop.

02

Part (a) Step 2 : Simplify

The magnetic field dipole moment M=8脳1022Am2current-carrying loop is just like a magnetic dipole whose magnetic dipole :

M=i脳Ai=MAi=8脳1022蟺1.5脳1062i=1.13脳1010A

03

Part (b) Step 1 : Given Information.

We need to find density J in the current loop.

04

Part (b) Step 2: Simplify

The current density J=?

So,

role="math" localid="1650965922291" i=J脳sJ=isJ=1.13脳1010蟺0.5脳1062J=0.014A/m2

05

Part (c) Step 1 : Given Information : 

We need to find the current density of a 1.0Acurrent in a 1.0mm-diameter wire.

06

Part (c) Step 2: Simplify

The wire with a diameter 1mm, the radius is r2=0.5mmand the current flows is I2=1A.

We finding the J2as:

J2=I2蟺r2=(1A)蟺0.5脳10-3m2=1.3脳106A/m2

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

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