The magnetic field 40.0 cm away from a long, straight wire...

The magnetic field $40.0 \mathrm{~cm}$ away from a long, straight wire carrying current $2.00 \mathrm{~A}$ is $1.00 \mu \mathrm{T}$. (a) At what distance is it $0.100 \mu \mathrm{T} ?$ (b) At one instant, the two conductors in a long household extension cord carry equal 2.00-A currents in opposite directions. The two wires are $3.00 \mathrm{~mm}$ apart. Find the magnetic field $40.0 \mathrm{~cm}$ away from the middle of the straight cord, in the plane of the two wires. (c) At what distance is it one-tenth as large? (d) The center wire in a coaxial cable carries current $2.00 \mathrm{~A}$ in one direction, and the sheath around it carries current $2.00 \mathrm{~A}$ in the opposite direction. What magnetic field does the cable create at points outside?

Key Concepts

Magnetic Field Cancellation in Coaxial and Parallel Conductor Arrangements

In configurations such as coaxial cables or closely spaced parallel wires with equal and opposite currents, the magnetic fields created by individual currents can cancel each other outside the conductor. This concept is important in practical applications where minimizing external magnetic fields is desired.

Superposition Principle for Magnetic Fields

In systems with multiple current-carrying conductors, the net magnetic field at a point is found by vectorially adding the individual magnetic fields produced by each conductor. This principle allows for the analysis of more complex arrangements where fields may reinforce or cancel each other.

Magnetic Field of a Long Straight Conductor

The magnetic field generated by a long straight conductor is given by the equation B = (?0I)/(2?r), indicating that the field is directly proportional to the current and inversely proportional to the distance from the wire. This formula is central to understanding how magnetic fields behave around current-carrying wires.

Ampere’s Law

Ampere’s Law relates the magnetic field around a closed loop to the total current passing through the loop. It is highly useful in systems exhibiting symmetry, providing a means to calculate the magnetic field generated by current-carrying conductors such as long, straight wires or solenoids.

Transcript

00:01 For this problem on the topic of the magnetic field, we are told that 40 centimeters from away from a long straight wire carrying current of two ampiers, the magnetic field is one microtteslas.

00:12 We want to know the distance at which it is 0 .1 microtteslas, and then we are told that at an instant, two conductors in a long extension cord carry two ampier currents in opposite direction.

00:25 If the two wires are 3mm apart, we want to find the magnetic field 40 centimeters away from the middle of the straight cord in the plane of the two wires.

00:36 We then want to find the distance at which it is 1 tenth as large, and then we are told that the central wire in a coaxial cable carries 2 amperes in one direction, and the surrounding sheath carries a current of 2 ampiers in the opposite direction.

00:52 We want to know what magnetic field this cable will create at any point outside.

00:59 So firstly, we know that the magnetic field b is equal to mu not i over 2 pi r.

01:08 So b is therefore proportional to 1 over r or inversely proportional to r.

01:14 And so the field will be one tenth as large at a 10 times larger distance, which means that r is equal to 400.

01:26 Centimeters or four meters.

01:35 Next for part b we are told that we have two conductors carrying current at equal at two ampiers in opposite directions and the wires are three millimeters apart.

01:49 We want to find the magnetic field 40 centimeters away from the middle of the straight cord...

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