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Question: A magnetic dipole is under the influence of two magnetic fields. The angle between the field directi...

A magnetic dipole is under the influence of two magnetic fields. The angle between the field directions is 60∘{60^ \circ }, and one of the fields has a magnitude of 1.2×10−2T1.2 \times {10^{ - 2}}T. If the dipole comes to stable equilibrium at an angle of 15∘{15^ \circ } with this field, what is the magnitude of the field?
A 3(3−1)×10−3T3\left( {\sqrt 3 - 1} \right) \times {10^{ - 3}}T
B (3−1)×10−3T\left( {\sqrt 3 - 1} \right) \times {10^{ - 3}}T
C 6(3−1)×10−3T6\left( {\sqrt 3 - 1} \right) \times {10^{ - 3}}T
D 2(3−1)×10−3T2\left( {\sqrt 3 - 1} \right) \times {10^{ - 3}}T

Explanation

Solution

When an electric current flows through a wire, a magnetic field can be created around it. Around the wire, concentric circles the magnetic field lines formed. The direction of the magnetic field depends on the direction of the current. It can be resolved by using the right-hand rule. The magnitude of the magnetic field depends on the distance from the charge-carrying wire and the amount of current. The formula includes the constant, that is the permeability of free space.
Formula used:
Magnetic field magnitude B=μ0I2πrB = \dfrac{{{\mu _0}I}}{{2\pi r}}, μ0={\mu _0} = the permeability of free space.
MB1sin⁡θ1=MB2sin⁡θ2M{B_1}\sin {\theta _1} = M{B_2}\sin {\theta _2}

Complete answer:
Let, Magnitude of one of the magnetic fields, B1=1.2×10−2T{B_1} = 1.2 \times {10^{ - 2}}T
The magnitude of the other magnetic field B2{B_2}
The angle between the two fields, θ=60∘\theta = {60^ \circ }.
A stable equilibrium angle between dipole and field {B_1}$$$${\theta _1} = {15^ \circ } .
The angle between dipole and field B2{B_2}, θ2=θ−θ1=60∘−15∘=45∘{\theta _2} = \theta - {\theta _1} = {60^ \circ } - {15^ \circ } = {45^ \circ }
The torque between both the poles balances each other.
The torque due to field {B_1}$$$$ = Torque due to field B2{B_2}
MB1sin⁡θ1=MB2sin⁡θ2M{B_1}\sin {\theta _1} = M{B_2}\sin {\theta _2} (M=M = Magnetic moment of the dipole)
Therefore, B2=B1sin⁡θ1sin⁡θ2{B_2} = \dfrac{{{B_1}\sin {\theta _1}}}{{\sin {\theta _2}}}
⇒B2=1.2×10−2×sin⁡15∘sin⁡45∘\Rightarrow {B_2} = 1.2 \times {10^{ - 2}} \times \dfrac{{\sin {{15}^ \circ }}}{{\sin 45 \circ }}
⇒B2=4.39×10−3T\Rightarrow {B_2} = 4.39 \times {10^{ - 3}}T
⇒B2≈3(3−1)×10−3T\Rightarrow {B_2} \approx 3\left( {\sqrt 3 - 1} \right) \times {10^{ - 3}}T

Hence, the correct answer is option A.

Note:
The unit of the magnetic field is Tesla.
The magnetic dipole moment is defined in terms of the torque in a magnetic field. In The same magnetic field, the larger torque creates the larger magnetic moment. The torque doesn't only depend on the magnitude of the magnetic moment but also the orientation relative to the magnetic field.