The conducting rod ab shown below makes frictionless contact with metal rails ca and db. The apparatus is in a uniform magnetic field of 0.855 T, perpendicular to the plane of the figure. The length L of the rod is 25 cm. (a) Find the magnitude of the emf induced in the rod when it is moving towards the right with a speed of 6.50 m/s. (b) In what direction does the current flow in the rod (clockwise or counter-clockwise)? Write several sentences that clearly explain your reasoning. (c) If the resistance of the circuit abdc is a constant 3.5 Ù, find the magnitude and direction of the force required to keep the rod moving to the right with a constant speed of 6.5 m/s.
To find the magnitude of the emf induced in the rod, you can use Faraday's law of electromagnetic induction. The formula for calculating the emf is given by:
emf = B * L * v
where B is the magnetic field strength, L is the length of the rod, and v is the velocity of the rod.
In this case, B = 0.855 T, L = 25 cm = 0.25 m, and v = 6.50 m/s. Plugging these values into the equation, you get:
emf = 0.855 T * 0.25 m * 6.50 m/s
emf ≈ 1.401 V
So the magnitude of the induced emf in the rod is approximately 1.401 V.
To determine the direction of the current flow in the rod, you can use the right-hand rule for magnetic fields and current. If you point your thumb in the direction of the velocity of the rod (to the right in this case), and your fingers in the direction of the magnetic field (perpendicular to the plane of the figure, into the page), then the direction your palm faces indicates the direction of the induced current. In this case, your palm would be facing downwards, which means the current flows clockwise.
Finally, to find the magnitude and direction of the force required to keep the rod moving to the right at a constant speed of 6.5 m/s, you can use the equation:
force = emf / resistance
Given that the resistance of the circuit abdc is 3.5 Ω, you can substitute the emf value from part (a) into the equation:
force = 1.401 V / 3.5 Ω
force ≈ 0.400 N
So the magnitude of the force required to keep the rod moving to the right at a constant speed is approximately 0.400 N. The direction of the force can be determined using the right-hand rule for current and magnetic fields. Since the current flows clockwise in the rod (as determined in part (b)), and the magnetic field is perpendicular to the plane of the figure (into the page), the force will be directed downward.