In Fig. 29-46, two concentric circular loops of wire...

In Fig. $29-46,$ two concentric circular loops of wire carrying current in the same direction lie in the same plane. Loop 1 has radius 1.50 $\mathrm{cm}$ and carries 4.00 $\mathrm{mA} .$ Loop 2 has radius 2.50 $\mathrm{cm}$ and carries 6.00 mA. Loop 2 is to be rotated about a diameter while the net magnetic field $\vec{B}$ set up by the two loops at their common center is measured. Through what angle must loop 2 be rotated so that the magnitude of that net field is 100 $\mathrm{nT}$ ?

In Fig. $29-46,$ two concentric circular loops of wire carrying
current in the same direction lie in
the same plane. Loop 1 has radius
1.50 $\mathrm{cm}$ and carries 4.00 $\mathrm{mA} .$ Loop 2
has radius 2.50 $\mathrm{cm}$ and carries 6.00 mA. Loop 2 is to be rotated about a diameter while the net magnetic field $\vec{B}$ set up by the two loops at their common center is
measured. Through what angle must loop 2 be rotated so that the
magnitude of that net field is 100 $\mathrm{nT}$ ?

Key Concepts

Vector Projection and Trigonometric Relationships in Field Analysis

Magnetic fields are vector quantities, and understanding their direction and magnitude involves breaking them into components. The cosine function arises naturally when projecting a vector onto an axis, which is essential when analyzing how rotating one of the sources affects the net magnetic field. This approach is widely used in various areas of physics to resolve the effects of vector rotation.

Effect of Orientation on Magnetic Field Contribution

When a current-carrying loop is rotated relative to a given direction, the effective contribution of its magnetic field in that direction is reduced. The magnitude of the component of the field along a particular axis is given by the original magnitude multiplied by the cosine of the angle between the field direction and that axis. This trigonometric dependence is central to understanding how the field changes with rotation.

Superposition Principle in Magnetism

In electromagnetism, the superposition principle states that the net magnetic field at a point due to multiple sources is the vector sum of the individual fields produced by each source independently. This concept is crucial when dealing with systems involving several current-carrying loops or coils, as it allows for the calculation of the overall field by combining the separate contributions.

Magnetic Field of a Circular Current Loop

A circular loop carrying an electric current produces a magnetic field along its axis. Using the Biot–Savart law, one can derive that the magnetic field at the center of the loop is proportional to the current and inversely proportional to the loop’s radius. This relation forms the basis for calculating the induced magnetic field in many electromagnetic applications.

Transcript

00:01 Hi there.

00:02 So for this problem, we need to calculate the angle that the loop 2 must be rotated so that the magnitude of the net field, magnetic field, is 100 nanostela.

00:19 So to do this, the first thing that we need to consider is the equation for the magnetic field, of a circular loop, in this case a wire that is carrying a current.

00:38 So that expression for a circular loop that is carrying a current, of course, is going to be of this form, where i is the current, the magnitude of the current, and arc is the radius of the circular loop.

01:13 So since we have in here two concentric loops, and so we will have two magnetic fields, one that is produced by the wire one and the other magnetic field that is produced by the wire two.

01:35 So for wire one, we will have this that is a count, the current that is different over the radius of that loop.

01:51 So on the other hand, we have for the loop 2 something similar but changing the values of the current and the radius.

02:05 So with this set, we can find these values because we are giving all of this information in the data given for this problem.

02:19 So we are going to do that.

02:22 Remember that you need to transform all of these values to the usual units.

02:34 So we will transform centimeters to meters and million pairs to one pairs.

02:41 So for b1 we have the magnitude of.

02:50 Of the constant mu .0, which is 1 .2 times 10 to the minus sets.

03:00 We also have the magnitude of the current, which is 4 times 10 to the minus 3.

03:10 And we also have this over the radius of the first loop, which is 0 .050.

03:21 So this gives a value of 167 .4 nanotesa.

03:32 And for the loop 2, we will have the same constant 1 .256 times 10 to minus 6.

03:45 The current changes, the current of the second loop is 6 million pairs, which is 6 times 10 to the minus 6.

03:53 And this over two times the radius of that loop which is 0 .0 .0 .0 .0 .0 .25 centimeters.

04:11 This loop is, as you can see, is bigger.

04:16 So it will be 150 .72.

04:25 Nanometers, nanotessla.

04:28 Sorry.

04:29 So these are the two expressions...

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