To determine the magnitude of the electromotive force (e.m.f) induced in a rectangular loop due to its motion in the magnetic field created by a long current-carrying wire, we can apply Faraday's law of electromagnetic induction. This law states that the induced e.m.f in a closed loop is equal to the negative rate of change of magnetic flux through the loop. Let's break this down step by step.
Understanding the Magnetic Field
First, we need to recognize the magnetic field generated by the infinitely long wire carrying current 'i'. The magnetic field (B) at a distance 'r' from a long straight wire is given by the formula:
B = (μ₀ * i) / (2π * r)
Here, μ₀ is the permeability of free space, approximately equal to 4π × 10⁻⁷ T·m/A. In our case, the distance 'r' from the wire to any point in the loop will change as the loop moves away from the wire.
Position of the Loop
Initially, the distance from the wire to the closest edge of the loop is 'x'. As the loop moves away with a velocity 'v', the distance from the wire to the closest edge of the loop becomes:
r = x + vt
where 't' is the time elapsed since the loop started moving. The magnetic field at this distance can be expressed as:
B(t) = (μ₀ * i) / (2π * (x + vt))
Calculating the Magnetic Flux
The magnetic flux (Φ) through the rectangular loop is the product of the magnetic field and the area of the loop that is exposed to this field. The area (A) of the loop is given by:
A = l * b
Thus, the magnetic flux through the loop at time 't' is:
Φ(t) = B(t) * A = (μ₀ * i * l * b) / (2π * (x + vt))
Finding the Induced e.m.f
According to Faraday's law, the induced e.m.f (ε) is the negative rate of change of magnetic flux:
ε = -dΦ/dt
To find this, we need to differentiate the magnetic flux with respect to time:
ε = -d/dt [(μ₀ * i * l * b) / (2π * (x + vt))]
Using the quotient rule for differentiation, we find:
ε = (μ₀ * i * l * b * v) / (2π * (x + vt)²)
Final Expression for e.m.f
Thus, the magnitude of the induced e.m.f in the loop as it moves away from the wire is given by:
ε = (μ₀ * i * l * b * v) / (2π * (x + vt)²)
This equation shows that the induced e.m.f depends on several factors: the current in the wire, the dimensions of the loop, the speed at which the loop moves away, and the distance from the wire. As the loop moves further away, the e.m.f decreases due to the increasing distance affecting the magnetic field strength.
In summary, by applying Faraday's law and understanding the relationship between magnetic fields and motion, we can derive the e.m.f induced in a moving rectangular loop near a current-carrying wire. This principle is fundamental in electromagnetic induction and has numerous applications in electrical engineering and technology.
