A rectangular wire loop has a 10 ohm resistor on the back as it moves into a region of uniform B=3T field at v=10m/s. The rectangle is w=5cm wide and l=20cm long. How much heat is generated in the resistor during the loop's entire path into and out of the magnetic field in
Joules?
To solve this problem, we need to use the formula for the heat generated in a resistor:
Heat (Q) = I^2 * R * t
Where:
- Q is the heat generated (in Joules)
- I is the current flowing through the resistor (in Amperes)
- R is the resistance of the resistor (in Ohms)
- t is the time interval (in seconds)
First, we need to find the value of the current flowing through the resistor. The current can be calculated using Ohm's Law:
I = V / R
Where:
- I is the current flowing through the resistor (in Amperes)
- V is the voltage across the resistor (in Volts)
- R is the resistance of the resistor (in Ohms)
In this case, the voltage across the resistor can be found using Faraday's Law of Electromagnetic Induction:
V = B * l * v
Where:
- V is the voltage across the resistor (in Volts)
- B is the magnetic field strength (in Tesla)
- l is the length of the wire (in meters)
- v is the velocity of the wire (in meters/second)
Now let's plug the numbers into the formulas:
B = 3 T
l = 0.2 m
v = 10 m/s
V = B * l * v
V = 3 * 0.2 * 10
V = 6 V
Finally, we can calculate the heat generated by substituting the values into the formula:
Q = I^2 * R * t
First, we need to find the time interval (t) it takes for the loop to move into and out of the magnetic field. Since the magnetic field is uniform, the time it takes to pass through the magnetic field is equal to the time it takes to cover the length of the wire.
t = l / v
t = 0.2 / 10
t = 0.02 s
Now we can substitute the values:
Q = (V / R)^2 * R * t
Q = (6 / 10)^2 * 10 * 0.02
Q = 0.36 * 10 * 0.02
Q = 0.072 J
Therefore, the heat generated in the resistor during the loop's entire path into and out of the magnetic field is 0.072 Joules.