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A block of mass m and charge q is released on a long smooth inclined plane magnetic field B is constant, uniform, horizontal, and parallel to the surface as shown. Find the time from the start when the block loses contact with the surface.
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(A)mcosθqB
(B)none
(C)msecθqB
(D)mcotθqB

Answer
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Hint: A free body diagram has to be drawn for the block showing the forces act on it. From the diagram, the force is found in terms of the mass of the block. Another force exists here due to the charge on a magnetic field. By equating these forces for the given condition the velocity of the block can be calculated. After getting the velocity the required time can also be found from the equation of motion of the block.

Formula used:
F1=mgcosθ
mg is the weight of the block and mgcosθ is the cosine component of mg
The force F1 is acting equal and opposite to the cosine component of the weight of the block.
The force occurs due to the magnetic field,
F2=qvB
B is the magnetic field and charge q
The velocity at time t
v=gsinθ.t

Complete answer:
The given diagram is,
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The diagram has to be modified to show the forces acting on it and also on the system.
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The weight of the block mg is acting normally downwards to the block and it is divided into two components mgcosθ and mgsinθat the angle of θ (angle of inclination).
The force F1 is acting equal and opposite to the cosine component of the weight of the block.
So, F1=mgcosθ
And due to the magnetic field B and charge q the force occurs, F2=qvB,
v is the velocity.
These two forces are equal when the block start leaving the incline plane having the effect of the magnetic field,
So, F1=F2
qvB=mgcosθ
v=mgcosθqB
Now the velocity of the block at time t ,
v=gsinθ.t
t=vgsinθ
put the value of v we get,
t=mgcosθqBgsinθ
t=mcosθqBsinθ
t=mcotθqB
So, the time from start when block loses contact with the surface t=mcotθqB
Hence, the right answer is in option (D).

Note:
The magnitude of the force on a magnetic field having a charge q is F=qvBsinθ where θ is the angle < 180 degrees between the velocity and the magnetic field. This shows that the magnetic force on a stationary charge or a charge moving parallel to the magnetic field is zero. if the direction of the velocity is normal to the magnetic field the force will beF=qvB. The direction of the force is given by the right hand rule.
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