Two magnetic dipoles X and Y are placed at a separation $d$, with their axes perpendicular to each other. The dipole moment of Y is twice that of X. A particle of charge $q$ is passing through their mid-point P, at angle $\theta=45^{\circ}$ with the horizontal line as shown in the figure. What would be the magnitude of force on the particle at that instant? ($d$ is much larger than the dimensions of the dipole)

(A) $\left(\frac{\mu_{0}}{4 \pi}\right) \frac{M}{(d / 2)^{3}} \times qv$

(C) $\left(\frac{\mu_{0}}{4 \pi}\right) \frac{2 M}{(d / 2)^{3}} \times qv$

(D) $\sqrt{2}\left(\frac{\mu_{0}}{4 \pi}\right) \frac{M}{(d / 2)^{3}} \times qv$

JEE Main · 2019 · 08 Apr 2019 · Shift 2 · Magnetic Effects of Current

Two magnetic dipoles X and Y are placed at a separation

d

, with their axes perpendicular to each other. The dipole moment of Y is twice that of X. A particle of charge

q

is passing through their mid-point P, at angle

θ = 4 5^{\circ}

with the horizontal line as shown in the figure. What would be the magnitude of force on the particle at that instant? (

d

is much larger than the dimensions of the dipole)

(A) $(\frac{μ _{0}}{4 π}) \frac{M}{( d /2 ) ^{3}} \times q v$
(B)0✓ Correct
(C) $(\frac{μ _{0}}{4 π}) \frac{2 M}{( d /2 ) ^{3}} \times q v$
(D) $2 (\frac{μ _{0}}{4 π}) \frac{M}{( d /2 ) ^{3}} \times q v$

Answer: (B)

Solution

Point P lies on the axial line of dipole X and on the equatorial line of dipole Y, both at a distance

d /2

. The axial field of X is

B_{1} = 2 (\frac{μ _{0}}{4 π}) \frac{M}{( d /2 ) ^{3}}

, and the equatorial field of Y, whose moment is

2 M

, is

B_{2} = (\frac{μ _{0}}{4 π}) \frac{2 M}{( d /2 ) ^{3}}

. These two are equal in size, and since the dipole axes are at right angles the fields are perpendicular to each other, so the net field has magnitude

B_{1}^{2} + B_{2}^{2} = 2 B_{1}

and points at

4 5^{\circ}

to the horizontal. The particle also moves at

4 5^{\circ}

to the horizontal, so its velocity is parallel to the net field. Since

F = q (v \times B)

vanishes when

v

and

B

are parallel, the force on the particle at that instant is zero.

JEE Main · 2019 · 08 Apr 2019 · Shift 2 · Single correct option

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JEE Main · 2019 · 08 Apr 2019 · Shift 2 · Magnetic Effects of Current

Solution

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