A rectangular loop of wire of mass m and resistance R slides on a friction-free horizontal table. There is a region of uniform magnetic field perpendicular to the table. What type of graph best describes the speed v of the loop as a function of time?
To determine the type of graph that best describes the speed of the loop as a function of time, we can analyze the forces acting on the loop.
When the loop moves through the uniform magnetic field, there will be an induced emf (electromotive force) according to Faraday's law of electromagnetic induction. This induced emf will create an electric current in the loop, which in turn produces a magnetic force that opposes the motion of the loop. This force is called the electromagnetic or Lorentz force.
The Lorentz force can be calculated using the equation:
F = BIL
Where F is the magnetic force, B is the magnetic field strength, I is the current flowing in the loop, and L is the length of the wire.
Since the loop is sliding on a friction-free horizontal table, the only force acting on the loop is the Lorentz force. According to Newton's second law, the net force on the loop is equal to the mass of the loop multiplied by its acceleration:
F = ma
Therefore, equating the Lorentz force and the net force, we have:
BIL = ma
The current flowing in the loop (I) can be calculated using Ohm's law: I = V/R, where V is the induced emf and R is the resistance.
Substituting I in the equation, we get:
BV/R = ma
Simplifying the equation, we obtain:
a = (B^2VL) / (Rm)
From this equation, we can see that the acceleration (a) of the loop is directly proportional to the magnetic field strength (B), the induced emf (V), and the length of the wire (L). It is inversely proportional to the resistance of the wire (R) and the mass of the loop (m).
Based on this analysis, we can deduce that the speed of the loop will increase with time. However, the rate of increase will depend on the factors mentioned above. Therefore, the graph that best describes the speed v of the loop as a function of time would be an increasing curve, but the specific shape would depend on the values of B, V, L, R, and m.