A coil with resistance R is placed in a magnetic field. The flux linked with the coil is ϕ . If the magnetic field suddenly reverses in direction, how much charge will passes through any cross section area of the coil

To determine how much charge passes through a cross-section of the coil when the magnetic field reverses, we can apply Faraday's law of electromagnetic induction and Ohm's law. Let's break this down step by step.

Understanding the Basics

When a magnetic field changes around a coil, it induces an electromotive force (EMF) in the coil. This phenomenon is described by Faraday's law, which states that the induced EMF is equal to the negative rate of change of magnetic flux through the coil:

Faraday's Law: EMF = -dϕ/dt

Here, ϕ represents the magnetic flux, which is the product of the magnetic field strength (B), the area of the coil (A), and the cosine of the angle (θ) between the magnetic field and the normal to the coil's surface:

Magnetic Flux: ϕ = B × A × cos(θ)

Calculating the Induced EMF

When the magnetic field reverses direction, the change in magnetic flux can be significant. If we assume the magnetic field changes from B to -B, the change in flux can be expressed as:

Change in Flux: Δϕ = ϕ_final - ϕ_initial = (-B × A) - (B × A) = -2B × A

Substituting this into Faraday's law gives us:

EMF = -dϕ/dt = -(-2B × A / Δt) = 2B × A / Δt

Using Ohm's Law to Find Charge

Now that we have the induced EMF, we can use Ohm's law to find the current (I) flowing through the coil:

Ohm's Law: I = EMF / R

Substituting the expression for EMF, we get:

I = (2B × A / Δt) / R = (2B × A) / (R × Δt)

Calculating the Total Charge

The total charge (Q) that passes through the coil in a time interval Δt can be calculated using the relationship between charge, current, and time:

Charge: Q = I × Δt

Substituting the expression for current, we find:

Q = ((2B × A) / (R × Δt)) × Δt = (2B × A) / R

Final Result

Thus, the total charge that passes through any cross-section of the coil when the magnetic field reverses direction is given by:

Q = (2B × A) / R

This formula shows that the amount of charge is directly proportional to the strength of the magnetic field and the area of the coil, while being inversely proportional to the resistance of the coil. This relationship highlights the interplay between magnetic fields and electrical circuits, a fundamental concept in electromagnetism.