infinitely long wire carries a current I100A. Below the wire a rod of length L 10cm is forced to move at a constant speed v=5m/s long horizontal conducting rails. The rod and rails form a conducting loop. The rod has resistance of R 0.4ohms. 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. What is the magnitude of the emf induced in the loop?
Let x be the distance from the right end of the rails to the rod.
Magnetic field of the long straight wire is
B=μ₀I/2πr
If the infinitesimal horizontal strip of length x and width dr, parallel to the wire and
a distance r from it => area A = x dr and the flux is
dΦ=BdA= (μ₀I/2πr)xdr
Φ=∫ dΦ= (μ₀Ix/2π)∫dr/r (limits: from ‘a’ to ‘a+L’) = (μ₀Ix/2π) ln{(a+L)/a}
ℰ=dΦ/dt = (μ₀I/2π) (dx/dt) ln{(a+L)/a} =(μ₀Iv/2π) ln{(a+L)/a}=
=(4π•10⁻⁷•100•5/2π)• ln{(1+10)/1}=10⁻⁴•ln11=2.4•10⁻⁴ V
To find the magnitude of the electromotive force (emf) induced in the loop, we can use Faraday's law of electromagnetic induction. According to Faraday's law, the emf induced in a conducting loop is equal to the rate of change of magnetic flux through the loop.
Here are the steps to calculate the magnitude of the emf induced in the loop:
1. Calculate the magnetic field at the position of the loop:
- The magnetic field due to an infinitely long wire carrying current is given by the formula: B = (µ₀ * I) / (2π * r), where µ₀ is the permeability of free space, I is the current, and r is the distance from the wire.
- In this case, the distance from the wire (r) is given as 10mm, which equals 0.01m.
- The current (I) is mentioned as I100A, which means 100A.
- The value of µ₀ is 4π × 10^(-7) T·m/A.
- Plugging in the values, we get B = (4π × 10^(-7) T·m/A * 100A) / (2π * 0.01m) = 0.02 T.
2. Calculate the magnetic flux through the loop:
- The magnetic flux is given by the formula: Φ = B * A, where B is the magnetic field and A is the area of the loop.
- In this case, the area (A) of the loop is equal to the product of the length (L) of the rod and the distance (a) from the wire.
- The length (L) is mentioned as 10cm, which equals 0.1m.
- The distance (a) from the wire is mentioned as 10mm, which equals 0.01m.
- The area (A) is therefore 0.1m * 0.01m = 0.001 m^2.
- Plugging in the values, we get Φ = 0.02 T * 0.001 m^2 = 0.00002 Wb.
3. Calculate the rate of change of magnetic flux:
- The rate of change of magnetic flux (dΦ/dt) represents how quickly the magnetic flux changes through the loop.
- In this case, the loop is moving at a constant speed (v) of 5m/s horizontally.
- As the loop moves, the distance (a) from the wire remains constant, but the length (L) of the rod that intersects the magnetic field changes.
- So, the rate of change of magnetic flux can be calculated as (dΦ/dt) = (dB/dt) * A, where (dB/dt) represents the rate of change of the magnetic field.
- Since the magnetic field (B) is constant, the rate of change of the magnetic field (dB/dt) is zero.
- Therefore, (dΦ/dt) = 0.
4. Calculate the emf induced in the loop:
- According to Faraday's law, the electromotive force (emf) induced in the loop is given by the formula: emf = -dΦ/dt.
- Since the rate of change of magnetic flux (dΦ/dt) is zero in this case, the magnitude of the emf induced in the loop is also zero.
Therefore, the magnitude of the emf induced in the loop is zero in this situation.