Torque on Current Loop

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When the current is passed through this loop, a magnetic field is produced which exerts a torque on the loops, rotating the shaft.

Here, the magnetic field is uniform all around it.

A current-carrying loop of wire in the above arrangement is attached to a vertical rotating shaft that feels magnetic forces that produce a clockwise torque as viewed from above.

This is how electrical energy is transformed into mechanical work.

Torque on a Current Loop in a Magnetic Field

If you look at Fig.1, four wires are joined to form a loop. They are placed inside the magnetic field.

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When the current is passed through this loop, the magnetic field of lines crosses the loop. It experiences a force, which in turn generates a torque due to the force as shown in Fig.2.

Now, the loop will experience a force due to the magnetic field.

Here, the current is flowing from D to C and the magnetic field in the opposite direction.

To find the direction of the force in the wire CD.

Let’s Apply Fleming’s Right-Hand Rule

It states that if we stretch our index finger, middle finger and the thumb in such a way that they are mutually perpendicular to each other where the index finger indicates the direction of the magnetic field, middle finger, the direction of an induced current, while the thumb represents the direction of motion.

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As we align our fingers in this way, we would observe that the current and the magnetic field are in opposite directions.

So, they are parallel, i.e. Sin 0° = 0.

Therefore, no force is acting on the wire CD. 

Similarly, if we look at wire AB, the direction of current is opposite to that in the wire CD, however in the same direction to that of the magnetic field.

Here also Sin 0° = 0, no force is acting on wire AB.

For wire CA,

The current is flowing in an upward direction, which means the electrons are flowing in the opposite direction. 

Applying Fleming’s right-hand rule, the magnetic field is in a direction perpendicular to that of current. The force is acting inwards.

While for the wire BD, the direction of the current is downward and the direction of force is outward.

A torque is exerted on the loop about an axis, making the loop rotate.

Now, we’ll deduce the equation for the same.

Torque on a Current Loop Equation

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Here, we would define some quantities:

1. W  = width of the loop

2. L = Height of the loop, which is the length of the wire feeling the force.

3. F = IBL 

We took F = IBL because in maximum cases, F and B are perpendicular to each other and Sin 90° = 1.

So, an equation for torque:

て= Fd  (d = distance of the wire from the axis of rotation)

d  = W/2

As F = IBL

So,  てleft  (torque from the left) = (IBL) W/2 and てright = (IBL) W/2, and when the loop is in the middle, no torque will act on it.

てnet =   てleft  + 0 +  てright  = BIHW

∵  HW = A (area of the loop)

て = BIA

Here, area vector A points outward in the middle.

So, て = IABSinӨ     

If there are N number of loops inside the field, we have:

て = NIABSinӨ      …(1)

Torque on Current Loop due to the Magnetic Moment

From equation (1),

Here, NIA is called the magnetic moment.

So, torque on any current-carrying loop is the magnetic moment times the magnetic field.

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Force is acting outwards whether the current is flowing inside or outside the loop, this magnetic force will continue spinning the loop.

Torque experienced by a Current Loop in a Uniform Magnetic Field

We know that the current loop when placed inside the magnetic field, behaves like a magnetic dipole where it has a North and the South pole.

So, magnetic moment, M = IA.

Now, consider a rectangular loop placed inside the magnetic field.

Explanation for Torque on Current Loop

Objects show certain movement or they exert a certain kind of force when pressure is used on them. For example, rotating a cap to open the bottle, removing the lid from a box, opening the door knob, tying the lace of the shoe and so on. These movements are torque movements which require some motion and are called rotational motions. Without the concept of a torque, there will be no movements at all. Torque is something that actually gives the rotational movement for all the objects without which we might not even be able to use these objects properly. 

The formula for torque is τ = F×r because torque is equal to the twisting force that tends to cause the movement or rotation. This formula is used when force (f) is applied to an object based on the distance ( r) between the center of rotation and to the point where force is applied. The direction of the torque can be found with the help of the right hand rule : where students have to curl their fingers on the right hand directed at the current and their thumb should stick out and point to the area vector. 

How to use Fleming’s Right Hand Thumb Rule 

To find out the direction of the torque, one can use the right hand thumb rule that was proposed by Fleming. By making sure that you are using the right formation, you can also find out the direction of the current. It is a common method to understand the directions and orientations of a three dimensional axis. We always have two possible orientations, so to see this:

• Hold hand outwards with palm facing up

• Curl fingers 

• Stick out thumb

This rule can also be used to find various other things like magnetic field, spirals, rotations, direction of the current and so on.