A conducting rod whose length is 25 cm is placed on a U-shaped metal wire that has a resistance R of 8 Ù. The wire and the rod are in the plane of the paper. A constant magnetic field of strength 0.4 T is applied perpendicular and into the paper. An applied force moves the rod to the right with a constant speed of 6 m/s.
(a)What is the magnitude of the induced emf in the wire?
(b) What is the magnitude and direction of the induced current in the wire?
To find the magnitude of the induced emf in the wire, we can use Faraday's Law of electromagnetic induction. The equation is given by:
Emf = -dΦ/dt,
where Emf is the induced emf, Φ is the magnetic flux, and dt is the change in time.
To calculate the induced emf, we need to find the magnetic flux passing through the wire. The magnetic flux is defined as the product of the magnetic field strength and the area through which it is passing. In this case, the area is given by the length of the rod times the width of the wire.
(a) The magnetic flux passing through the wire is given by:
Φ = B * A,
where B is the magnetic field strength and A is the area.
Given that the length of the rod is 25 cm (or 0.25 m), the width of the wire can be assumed to be negligible.
Therefore, A = length * width = 0.25 m * 0 = 0 m².
Since the area is 0 m², the magnetic flux through the wire is also 0.
Therefore, the magnitude of the induced emf in the wire is 0 V.
(b) Since the induced emf is 0 V, no current is induced in the wire.
To find the answers to these questions, we need to use Faraday's law of electromagnetic induction, which states that the magnitude of the induced emf (ε) is equal to the rate of change of magnetic flux (Φ) through a circuit. The magnetic flux is given by the product of the magnetic field strength (B) and the area (A) enclosed by the circuit.
(a) To find the magnitude of the induced emf, we first need to calculate the rate of change of magnetic flux. The area enclosed by the circuit is equal to the product of the length of the conducting rod (l) and the distance it moves (Δx). In this case, the rod moves with a speed of 6 m/s, and the length of the rod is 25 cm (which is equal to 0.25 m). Therefore, the area is A = l * Δx = 0.25 m * 6 m/s = 1.5 m^2.
Next, we need to calculate the rate of change of magnetic flux. Since the magnetic field is perpendicular to the area, the rate of change of magnetic flux is equal to the product of the magnetic field strength and the rate of change of the area. In this case, the magnetic field strength is 0.4 T, and the rate of change of the area is equal to the speed of the rod. Therefore, the rate of change of magnetic flux (dΦ/dt) is 0.4 T * 6 m/s = 2.4 T · m^2/s.
Finally, we can calculate the magnitude of the induced emf using Faraday's law: ε = -dΦ/dt = -2.4 T · m^2/s. The negative sign indicates that the induced emf opposes the change in magnetic flux.
(b) To find the magnitude and direction of the induced current, we need to use Ohm's law, which states that the current (I) is equal to the ratio of the emf (ε) to the resistance (R).
In this case, the resistance of the wire is given as 8 Ω. Therefore, the magnitude of the induced current (I) is equal to ε/R = (-2.4 T · m^2/s) / (8 Ω) = -0.3 A. The negative sign indicates that the direction of the induced current is opposite to the direction of the applied force that moves the rod to the right.
So, the magnitude of the induced emf is 2.4 T · m^2/s, and the magnitude of the induced current is 0.3 A, flowing in the opposite direction to the applied force.
You have formulas for induced EMF, and you need to use those.
However, if one considers the resistance constant, even if the length of the wire is changing, here is an alternative approach.
power= EMF^2/R
force*velocity=EMF^2/R
EMF= sqrt (force*velcity*resistance)
and current= sqrt (force*velocty/R)