A metal crossbar of mass m rides on a parallel pair of long horizontal conductin

Question

A metal crossbar of mass m rides on a parallel pair of long horizontal conducting rails separated by a distance L and connected to a device that supplies constant current I to the circuit, as shown below. The circuit is in a region with a uniform magnetic field B whose direction is vertically downward. There is no friction and the bar starts from rest at t = 0. Assume that the rails are frictionless but titled upward so that they make an angle theta with the horizontal, and with the current source attached to the low end of the rails. (a) What minimum value of B is needed to keep the bar from sliding down the rails? (Use the following as necessary: m, g, I, theta, and l.) |B| = (b) What is the acceleration of the bar if B is twice the value found in part (a) ? (Use the following as necessary: g and theta.) magnitude a = direction

Explanation / Answer

Here ,

a) for rhe rod to keep bar sliding down the rails

net force on the rod along the plane is zero

m* g * sin(theta) - B * v * L * cos(theta) = 0

B = m * g * tan(theta)/(v * L)

the minimum value of magnetic field needed is m * g * tan(theta)/(v * L)

b)

for value of magnetic field twice the found value

net force on the rod will be

Fnet = - m* g * sin(theta) + 2 * B * v * L * cos(theta)

putting value of B from a

Fnet = - m* g * sin(theta) + 2 * (m * g * tan(theta)/(v * L)) * v * L * cos(theta)

using second law of motion

m * a = m * g * sin(theta)

a = g * sin(theta)

the acceleration of the rod is g * sin(theta)

the direction of acceleration is upwards along the plane

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