# Wire loop in magnetic field of the long magnet

• sergiokapone
In summary, the current in the wire loop is flow along the sides of the circuit, and the torque is due to the ampere force acting on the sides of the wire loop.
sergiokapone

## Homework Statement

The wire loop ##ABCD## is in a magnetic field of the long thin rod magnet with a magnetic moment per unit volume of ##\mathfrak{M}_0## and a cross section ##S##. The north pole of this magnet is in the center of the loop, and the magnet is perpendicular to the plane of the loop. Emf is connected to the opposite edges of the diagonal ##AC## , thus the current I is flow along sides of the circuit. Find the torque of force couple M and its direction.

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## The Attempt at a Solution

The magnet field of the long magnet near the north pole one can represent as magnetic field of monopole

H = \frac{q_m}{r^2}

where ##q_m## -- effective magnetic charge, which one can find from definition of magnetic momentum ##p_m = q_ml##:
\begin{equation*}
q_ml = \mathfrak{M}_0 S l
\end{equation*}
thus, the effective magnetic charge is

q_m = \mathfrak{M}_0 S.

The Ampere Force acting on sides of wire loop:

dF = Idl H \sin\alpha,

And, so the torque is

M = \int rdF = I\mathfrak{M}_0 S \int \frac{\sin\alpha}{r} dl

from the geometry ##dl\sin\alpha = r d\alpha##, thus

M = \int rdF = I\mathfrak{M}_0 S 4 \frac{\pi}{2}

The answer in the problems book is

M = \int rdF = 4 I\mathfrak{M}_0 S

Where did I go wrong?

I believe you haven't taken into account the direction of the torque, ##\vec{dM} = \vec{r} \times \vec{dF}##, for each element of current ##dl##.

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Illustration for solution

Last edited:
sergiokapone said:
That's right, but what TS says is that not all ##\vec {dM}## are pointing in the same direction !

Ok,I lost sight of that, thanks. In the process of integration one will have to take into account the changes in the angles between the current and the radius vector, and count the direction of torque also. Tedious way.

Is there a more general way? (I mean using only vectors)
May be, an easier way:

\vec M = I \oint \vec{dl} (\vec B \cdot \vec r) - \vec B (\vec r \cdot \vec{dl}).

But following using Stokes' theorem ##I \oint_L \vec{dl} =I \int_S \vec{dS} \times \nabla## I get zero.

Last edited:
If you look at all the torque elements, you can see that the net torque vector is in a certain direction. So, you can just sum the torque components in that direction.

#### Attachments

• Torque on current square 2.png
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## 1. What is a wire loop in a magnetic field?

A wire loop in a magnetic field refers to a closed circuit made up of a conducting material, such as copper wire, placed within the influence of a magnetic field. When the loop is moved within the magnetic field, an electric current is induced within the wire due to the interaction between the magnetic field and the moving charges in the conductor.

## 2. How does a wire loop behave in a magnetic field?

A wire loop in a magnetic field experiences a force known as the Lorentz force, which is perpendicular to both the direction of the current and the direction of the magnetic field. This force causes the wire loop to move in a circular motion, with the direction of rotation determined by the direction of the current and the orientation of the magnetic field.

## 3. What is the purpose of a wire loop in a magnetic field?

A wire loop in a magnetic field can be used in various applications such as generators, motors, and sensors. In generators and motors, the wire loop is used to convert mechanical energy into electrical energy or vice versa. In sensors, the wire loop can be used to detect changes in the magnetic field, which can indicate the presence of an object or a change in position.

## 4. How does the strength of the magnetic field affect a wire loop?

The strength of the magnetic field has a direct impact on the magnitude of the induced current in a wire loop. A stronger magnetic field will induce a larger current, while a weaker magnetic field will induce a smaller current. This relationship is described by Faraday's law of induction.

## 5. What factors can affect the behavior of a wire loop in a magnetic field?

The behavior of a wire loop in a magnetic field can be affected by several factors, including the strength and orientation of the magnetic field, the speed of the wire loop within the field, the size and shape of the loop, and the resistance of the wire. These factors can impact the magnitude and direction of the induced current, as well as the overall motion of the wire loop.

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