Test: Torque on Current Carrying Loop - NEET MCQ

A current (i) in a magnetic field ( B) experiences a force ( F) given by the equation

 F = i (l × B) 

or F = ilB sin θ,

where l is the length of the wire, represented by a vector pointing in the direction of the current.

The magnetic moment of this loop is in direction perpendicular to the plane of loop and coming out of the plane. The magnetic field is also directed perpendicularly into the loop. So, magnetic moment magnetic field are antiparallel. The torque acting on the loop, which is given by the cross product of moment and field, is zero since sin 180 = 0

Concept:

• A coil always act as a magnetic dipole of dipole moment M.

The dipole moment of a coil is given by:

Dipole moment = M = NIA

Where N is number of turns, I is current and A is area of the coil

Area vector (A):

• The vector perpendicular to the plane of the coil is called as area vector.
• The magnitude of area vector is equal to area of the coil.

Torque of magnetic field on any magnetic dipole moment is given by;

Torque (τ) = M × B = MB Sinθ

Where M × B is cross product of Area vector A and magnetic field vector B and θ is the angle between area vector A and magnetic field B.

Now,

Given that:

Area = A = 2 × 4 = 8 m²

As the area is perpendicular to coil so angle between area vector and magnetic field B is 90°.

Angle = θ = 90°

Number of turns = N =100

Current = I = 3 A

Magnetic field = B = 0.05 T

Magnetic moment = M = NIA = 100 × 3 × 8 = 2400

Torque = τ = MB Sin (90°) = 2400 × 0.05 × 1= 120 Nm

The torque on a current loop depends upon the area of the current loop, when the magnetic field is perpendicular to the plane of the loops the torque has its maximum value,
Τ=I A B
We know I and B for all these cases but A depends upon the geometry. The circle has the greatest area so it should provide the greatest torque.

Torque:

• ​ It is defined as the product of the magnitude of the force and the perpendicular distance of the line of action of a force from the axis of rotation.
• Formula, τ = rFsinθ , where r = perpendicular distance, F = force, θ = angle between force and perpendicular distance
• The SI unit of torque is N-m.

Torque in a current loop:

• ​Let us consider a rectangular loop such that it carries a current of magnitude I.

• If we place this loop in a magnetic field, it experiences a torque but no net force, quite similar to what an electric dipole experiences in a uniform electric field.
• The torque acting on the coil is calculated as, τ = NIAB sinθ where N = number of turns, I = current, A = area of the loop, B = magnetic field, θ = angle between area and magnetic field. 

Calculation:

Given,

The number of turns, N = 100 turns

The current in the coil, I = 80 mA

Angle, θ = 30º

The radius of the circular coil, r = 0.50 m

The magnitude of the magnetic field, B = 4.0 T

The torque acting on the coil is calculated as, τ = NIAB sinθ 

τ = 100 × 80 × 10^-3 × π × (0.50)² × 4.0 × sin 30º

τ = 4π N m

Hence, the magnitude of torque acting on the coil is 4π N m.

Step 1: Given data
Area of the current loop, A = 0.01 m²
Electric current flowing in the current loop, I = 100 A
Magnetic field intensity, B = 0.1 T
Angle between the current loop and the magnetic field, θ = 0^∘
Torque acting on the current loop, τ = ?

Step 2: Formula used

τ = I A B sinθ…(a)

Step 3: Calculation of the torque acting on the current-carrying loop
Substituting the given values in equation (a), we get:

τ = 100 A × 0.01 m² × 0.1 T×sin⁡0^∘

τ = 100 × 0.01 × 0.1 × 0

τ = 0

The correct answer is option b) i.e. 

• Torque on a current loop: A current-carrying loop placed in a magnetic field will experience multiple magnetic forces on it due to the change in direction of current along the loop. Any two such equal and opposite force forms a couple and causes torque in the loop.
  • The torque experienced by a current-carrying loop placed in a magnetic field is the vector product of the magnetic moment and the magnetic field.

It is given by:

Torque, τ = m × B

Where m is the magnetic moment and B is the magnetic field intensity.

• The magnetic moment is the product of the current flowing in the loop and the area of the loop.

It is given by:

Magnetic moment, m = IA

Where I is the current and A is the area.

The torque acting on the loop, τ = m × B = IAB

• Since the loop is circular, area A = πr² (where r is the radius of the circular loop)

⇒ τ = IB(πr²)

⇒ τ ∝ r2

Torque acting on an equilateral triangle in a magnetic field B is
τ = I A B sinθ
Area of ΔLMN: 
and θ = 90^o
Substituting the given values in the expression for torque, we have

Torque on a rectangular current loop in a uniform magnetic field:

• If a rectangular loop carrying a steady current is placed in a uniform magnetic field then it will experience a torque.
  • The net force on the loop will be zero.
• The torque on the current-carrying rectangular loop is given as,

⇒ τ = NIAB.sinθ

Where N = number of turns in the coil,
I = current in the loop,
A = area enclosed by the loop,
B = magnetic field intensity, and 
θ = angle between the normal to the plane of the coil and the direction of a uniform magnetic field

The magnetic moment m for the loop is given as,

⇒ m = NIA

So,

We know that the magnetic moment 'm' of the current-carrying loop is given as,

⇒ m = NIA

Hence, option 1 is correct.

CONCEPT:

• As the current-carrying conductor experiences a force when placed in a magnetic field, each side of a current-carrying coil experiences a force in a magnetic field.
•  In the present section, we shall see in what way the loop carrying current is influenced by a magnetic field.

• Here θ be the angle between the plane of the rectangular coil and the magnetic field.
• Force acting on the current-carrying loop:

⇒ F = I L B sin θ,

Where F is force, I = current, L = distance from the axis, B = strength of the magnetic field, θ = angle between the plane of the rectangular coil and the magnetic field.

• Acting on the upper and lower sides are equal and opposite along the same line of action, they cancel each other.
• As the force acting on the sides QR and SP are equal and opposite along different lines of action they constitute a couple.
• So, the torque acting on the loop is,

⇒ T = force × arm of the couple

⇒ T = B I L × b Sin θ = B I A Sin θ

⇒ Torque (T) = B I A Sin θ

• The potential energy of the coil in the magnetic field is given by:

Potential energy (PE) = B I A Cos θ

Now,

⇒ Torque (T) = B I A Sin θ

⇒ Potential energy (PE) = B I A Cos θ

• Initially the magnetic field was perpendicular to the plane of the coil then angle between the area A and the magnetic field B is 0°.
  • So the torque will be zero in this case and the magnetic potential energy will be B I A which is maximum value.
• After turning, the angle between area A and the magnetic field is 90°.
  • So the magnetic torque will be maximum in this case and potential energy will be zero. So option 2 is correct.