A 0.330 kg bar is placed on a pair of frictionless rails separated by a distance of 1.60 m and located on an inclined plane that makes an angle of 29.0° with respect to the ground. The total resistance of the circuit remains at a constant 1.8 Ω and a uniform magnetic field of magnitude 0.510 T is directed downward, perpendicular to the ground, over the entire region through which the bar moves.
a) Once the bar reaches its terminal velocity, what must the magnitude of the
magnetic force be?
b) Once the bar reaches its terminal velocity, what is the magnitude of the induced current through the circuit?
c)Once the bar reaches its terminal velocity, what is the magnitude of the induced emf through the circuit?
d)What is the terminal velocity of the bar?
To solve this problem, we need to use the concepts of magnetic force, induced current, induced emf, and terminal velocity. Let's break down each part of the question and see how we can find the answers.
a) Once the bar reaches its terminal velocity, the net force acting on the bar will be zero. The forces acting on the bar in this situation are the gravitational force (mg) and the magnetic force (F_mag). The gravitational force can be calculated using the formula mg = m * g, where m is the mass of the bar and g is the acceleration due to gravity (approximately 9.8 m/s^2).
To find the magnetic force, we can use the formula F_mag = q * v * B, where q is the charge of the bar (in this case, q = 0 because the bar is neutral), v is the velocity of the bar, and B is the magnitude of the magnetic field. Since the bar is moving at its terminal velocity, the magnetic force will be equal in magnitude but opposite in direction to the gravitational force. Thus, F_mag = mg.
b) Once the bar reaches its terminal velocity, the induced current through the circuit will be zero. This is because the magnetic force acting on the bar is balanced by the resistance of the circuit, causing the net force on the bar to become zero. Consequently, there is no relative motion between the bar and the magnetic field, which means there is no induced current.
c) Once the bar reaches its terminal velocity, the magnitude of the induced emf (electromotive force) through the circuit will also be zero. Induced emf is determined by the rate of change of magnetic flux (Φ) through the circuit. When the bar is at rest, there is no change in magnetic flux, resulting in zero induced emf.
d) To calculate the terminal velocity of the bar, we can use the concept of balance of forces. At terminal velocity, the net force acting on the bar is zero. The forces acting on the bar are the gravitational force, the magnetic force, and the force of friction. The force of friction can be ignored in this case since the rails are assumed to be frictionless.
Since the bar is on an inclined plane making an angle of 29.0°, the perpendicular component of the gravitational force will be mg * sin(29.0°), and the parallel component of the gravitational force will be mg * cos(29.0°). These components will balance the magnetic force acting upwards.
mg * cos(29.0°) = mg * sin(29.0°)
g * cos(29.0°) = g * sin(29.0°)
cos(29.0°) = sin(29.0°)
Therefore, the terminal velocity of the bar will be when the bar is at rest since the forces are balanced.
In summary:
a) The magnitude of the magnetic force is equal to the gravitational force, which is mg.
b) The induced current through the circuit is zero when the bar reaches its terminal velocity.
c) The magnitude of the induced emf through the circuit is zero when the bar reaches its terminal velocity.
d) The terminal velocity of the bar is zero since the forces acting on it are balanced.