An infinitely long wire carries a current I=100A. Below the wire a rod of length L=10cm is forced to move at a constant speed v=5m/s along horizontal conducting rails. The rod and rails form a conducting loop. The rod has resistance of R=0.4Ω. The rails have neglibible resistance. The rod and rails are a distance a=10mm from the wire and in its non-uniform magnetic field as shown.
#8 (6 points possible)
What is the magnitude of the emf induced in the loop in V?
What is the magnitude of the emf induced in the loop in V? - unanswered
To find the magnitude of the emf induced in the loop, we can use Faraday's law of electromagnetic induction. According to Faraday's law, the emf induced in the loop is directly proportional to the rate of change of magnetic flux through the loop.
The formula for the emf induced in a loop is given by:
emf = -N * (dΦ/dt)
Where:
emf is the electromotive force (induced emf)
N is the number of turns in the loop
(dΦ/dt) is the rate of change of magnetic flux through the loop
In this case, the loop consists of a rod moving at a constant speed v along horizontal conducting rails. The magnetic field created by the infinitely long wire will cut across the loop, inducing an emf. The magnetic field is non-uniform as shown.
Given:
Current I = 100A (in the wire)
Length L = 10cm (length of the rod)
Speed v = 5m/s (velocity of the rod)
Resistance R = 0.4Ω (resistance of the rod)
Distance a = 10mm (distance between the rod and wire)
To find the magnitude of the emf induced in the loop, we need to calculate the rate of change of magnetic flux through the loop.
Let's break the problem down into steps:
Step 1: Calculate the magnetic field at the location of the rod.
The magnetic field produced by an infinitely long wire at a distance a from the wire is given by:
B = (μ0 * I) / (2π * a)
Where:
B is the magnetic field
μ0 is the permeability of free space (μ0 = 4π x 10^-7 Tm/A)
I is the current in the wire
a is the distance between the wire and the rod
Step 2: Calculate the area of the loop.
The area of the loop is simply the length of the rod multiplied by the distance between the rails (width of the loop). In this case, the width of the loop is negligible compared to the length of the rod, so we can approximate the area as:
A = L * w
Where:
A is the area of the loop
L is the length of the rod
w is the width of the loop (width of the rails)
Step 3: Calculate the rate of change of magnetic flux.
The rate of change of magnetic flux through the loop is given by:
(dΦ/dt) = B * v * cosθ
Where:
(dΦ/dt) is the rate of change of magnetic flux
B is the magnetic field
v is the velocity of the rod
θ is the angle between the magnetic field and the direction of motion of the rod
Step 4: Calculate the number of turns in the loop.
Since we only have one loop, the number of turns N is 1.
Step 5: Calculate the emf induced in the loop.
Finally, we can substitute the values we calculated into the formula for the emf:
emf = -N * (dΦ/dt)
Solving this equation will give us the magnitude of the emf induced in the loop.