Question

A voltmeter is being used to measure a 60 [Hz] voltage across a resistor in a circuit that is inside of a lab room. The two leads going from the voltmeter to the resistor are 5 [cm] apart and 2 [m] long, forming a rectangular loop with the voltmeter on one side and the resistor on the other side. Inside the lab there is a 60 [Hz] magnetic field coming from some equipment in the room. The magnetic flux density vector coming from this equipment has a peak magnitude of B0 = 0.01 [T] and the direction of the magnetic field vector from the equipment is at an angle of 45 from the normal to the rectangular loop formed by the leads. Calculate the peak magnitude of the extra (error) AC voltage that the voltmeter will read that is in addition to what the voltmeter is supposed to be reading.

Answer 1

The student is asking for the calculation of peak induced AC voltage in a loop caused by a nearby alternating magnetic field. Using Faraday's law and the provided parameters, such as the peak magnetic flux density, area of the loop, and angle of the magnetic field, one can find the peak induced voltage by the changing magnetic flux.

Step-by-step explanation:

The student is asking about the induced voltage (also known as electromotive force or emf) that a voltmeter would erroneously read due to a changing magnetic flux. The scenario involves a 60 Hz magnetic field intersecting a loop formed by voltmeter leads and a resistor, with the magnetic field at a 45-degree angle to the normal of the loop. To find the peak magnitude of the extra AC voltage the voltmeter reads, we use Faraday's law of electromagnetic induction, which states that:

E = -N * (dΦ/dt)
Where E is the induced emf, N is the number of turns (which is 1 in this case), and dΦ/dt is the rate of change of magnetic flux Φ through the loop. In our case, the magnetic flux is changing due to the alternating magnetic field, and the peak flux Φ can be found using:

Φ = B * A * cos(θ)
Given that B0 is the peak magnetic flux density (0.01 T), A is the area of the loop (0.1 m * 0.05 m = 0.005 m2), and θ is the angle between the magnetic field and the normal to the loop (45 degrees), we can calculate the peak flux. Since the magnetic field is alternating at 60 Hz, the rate of change of the flux will also be sinusoidal and can be found using the derivative of the sine function multiplied by the angular frequency. Finally, we substitute these values into Faraday's law to calculate the peak induced voltage.