1
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 10th January Morning Slot

An insulating thin rod of length $$l$$ has a linear charge density $$\rho \left( x \right)$$ = $${\rho _0}{x \over l}$$ on it. The rod is rotated about an axis passing through the origin (x = 0) and perpendicular to the rod. If the rod makes n rotations per second, then the time averaged magnetic moment of the rod is -
A
$${\pi \over 3}n\rho {l^3}$$
B
$${\pi \over 4}n\rho {l^3}$$
C
$$n\rho {l^3}$$
D
$$\pi n\rho {l^3}$$

Explanation

$$ \because $$   M = NIA

dq = $$\lambda $$dx  &   A = $$\pi $$x2

$$\int {dm} = \int {\left( x \right){{{\rho _0}x} \over \ell }} \,dx.\pi {x^2}$$

M = $${{n{\rho _0}\pi } \over \ell }.\int\limits_0^\ell {{x^3}.dx} = {{n{\rho _0}\pi } \over \ell }.\left[ {{{{L^4}} \over 4}} \right]$$

M = $${{n{\rho _0}\pi {\ell ^3}} \over 4}$$ or $${\pi \over 4}n\rho {\ell ^3}$$
2
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 10th January Evening Slot

At some location on earth the horizontal component of earth’s magnetic field is 18 × 10–6 T. At this location, magnetic needle of length 0.12 m and pole strength 1.8 Am is suspended from its mid-point using a thread, it makes 45o angle with horizontal in equilibrium. To keep this needle horizontal, the vertical force that should be applied at one of its ends is -
A
3.6 $$ \times $$ 10$$-$$5 N
B
1.8 $$ \times $$ 10$$-$$5 N
C
1.3 $$ \times $$ 10$$-$$5 N
D
6.5 $$ \times $$ 10$$-$$5 N

Explanation


$$\mu $$Bsin45o = F$${\ell \over 2}$$sin45o

F = 2$$\mu $$B
3
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 10th January Evening Slot

A hoop and a solid cylinder of same mass and radius are made of a permanent magnetic material with their magnetic moment parallel to their respective axes. But the magnetic moment of hoop is twice of solid cylinder. They are placed in a uniform magnetic field in such a manner that their magnetic moments make a small angle with the field. If the oscillation periods of hoop and cylinder are Th and Tc respectively, then -
A
Th = 1.5 Tc
B
Th = Tc
C
Th = 2Tc
D
Th = 0.5 Tc

Explanation

T = $$2\pi \sqrt {{1 \over {\mu B}}} $$

Th = $$2\pi \sqrt {{{m{R^2}} \over {\left( {2\mu } \right)B}}} $$

TC = $$2\pi \sqrt {{{1/2m{R^2}} \over {\mu B}}} $$
4
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 11th January Morning Slot

In an experiment, electrons are accelerated, from rest, by applying a voltage of 500 V. Calculate the radius of the path if a magnetic field 100 mT is then applied. [Charge of the electron = 1.6 $$ \times $$ 10–19 C Mass of the electron = 9.1 $$ \times $$ 10–31 kg]
A
7.5 $$ \times $$ 10$$-$$4 m
B
7.5 $$ \times $$ 10$$-$$3 m
C
7.5 m
D
7.5 $$ \times $$ 10$$-$$2 m

Explanation

$$r = {{\sqrt {2mk} } \over {eB}} = {{\sqrt {2me\Delta v} } \over {eB}}$$

$$r = {{\sqrt {{{2m} \over e}.\Delta v} } \over B} = {{\sqrt {{{2 \times 9.1 \times {{10}^{ - 31}}} \over {1.6 \times {{10}^{ - 19}}}}\left( {500} \right)} } \over {100 \times {{10}^{ - 3}}}}$$

$$r = {{\sqrt {{{9.1} \over {0.16}} \times {{10}^{ - 10}}} } \over {{{10}^{ - 1}}}} = {3 \over 4} \times {10^{ - 4}}$$

   $$ = 7.5 \times {10^{ - 4}}$$

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