Suppose that the light bulb in the figure is a 61.8-W bulb with a resistance of 218 Ω. The magnetic field has a magnitude of 0.37 T, and the length of the rod is 0.59 m. The only resistance in the circuit is that due to the bulb. What is the shortest distance along the rails that the rod would have to slide for the bulb to remain lit for one-half second?
Answer in m
To determine the shortest distance along the rails that the rod would have to slide for the bulb to remain lit for one-half second, we need to consider the interaction between the magnetic field and the moving rod.
The formula that relates the electrical power dissipated in a resistor to its resistance and the current passing through it is:
P = I^2 * R
Where P is the power (in watts), I is the current (in amperes), and R is the resistance (in ohms).
In this case, the power dissipated in the light bulb is given as 61.8 W and the resistance of the bulb is 218 Ω. We can use these values to calculate the current passing through the bulb.
Rearranging the formula, we have:
I = sqrt(P / R)
I = sqrt(61.8 W / 218 Ω)
I ≈ 0.47 A
Now, since the bulb needs to remain lit for one-half second, we can calculate the total charge passing through the rod during this time. The relationship between current, charge, and time is given by:
Q = I * t
Where Q is the charge (in coulombs), I is the current (in amperes), and t is the time (in seconds).
In this case, the time is given as 0.5 seconds. Substituting the values:
Q = 0.47 A * 0.5 s
Q = 0.235 C
The motion of the rod in the magnetic field induces an electromotive force (EMF) along the length of the rod, based on Faraday's law of electromagnetic induction:
EMF = B * L * v
Where EMF is the electromotive force (in volts), B is the magnetic field strength (in teslas), L is the length of the rod (in meters), and v is the velocity of the rod (in meters per second).
In this case, the magnetic field strength is given as 0.37 T and the length of the rod is given as 0.59 m.
We want to find the minimum distance d that the rod needs to slide, so we need to determine the velocity v in terms of d. The time of 0.5 seconds can be expressed as:
t = d / v
Rearranging the equation, we have:
v = d / t
Plugging in the values and rearranging:
0.47 A = (0.37 T * 0.59 m * d) / (0.5 s)
Solving for d, we get:
d = (0.47 A * 0.5 s) / (0.37 T * 0.59 m)
d ≈ 0.68 m
Therefore, the shortest distance along the rails that the rod would have to slide for the bulb to remain lit for one-half second is approximately 0.68 meters.