**Title: Maximum Potential Difference Induced Across a Simple Pendulum**

**Introduction:**
A simple pendulum consists of a mass (bob) attached to a conducting wire of length L, which is free to swing under the influence of gravity. When the pendulum swings through an angle 2θ, a potential difference is induced across it due to the Earth's magnetic field component B. In this explanation, we will explore the maximum potential difference induced across the pendulum and provide a detailed explanation.

**Understanding the Induced Potential Difference:**
- When a conductor moves in a magnetic field, an emf (electromotive force) is induced across it according to Faraday's law of electromagnetic induction.
- In this case, the conducting wire of the pendulum acts as the moving conductor, and the Earth's magnetic field component acts as the magnetic field.
- As the pendulum swings through an angle 2θ, the length of the wire moving perpendicular to the magnetic field changes, resulting in an induced emf.

**Calculating the Maximum Potential Difference:**
- The maximum potential difference induced across the pendulum can be calculated using the formula:
ΔV = B * L * v * sin(θ)
where ΔV is the potential difference, B is the Earth's magnetic field component, L is the length of the conducting wire, v is the velocity of the pendulum, and θ is the angle of swing.

**Deriving the Maximum Potential Difference Formula:**
- When the pendulum reaches its maximum displacement, all the potential energy is converted into kinetic energy.
- At this point, the velocity of the pendulum can be calculated using the conservation of mechanical energy formula:
m * g * L * (1 - cos(θ)) = 0.5 * m * v^2
where m is the mass of the bob, g is the acceleration due to gravity, L is the length of the conducting wire, θ is the angle of swing, and v is the velocity of the pendulum.
- Solving the above equation for v, we get:
v = sqrt(2 * g * L * (1 - cos(θ)))

**Substituting the Velocity in the Potential Difference Formula:**
- Substituting the expression for v in the potential difference formula, we get:
ΔV = B * L * sqrt(2 * g * L * (1 - cos(θ))) * sin(θ)
- Simplifying this equation further, we can express it as:
ΔV = B * L * sqrt(2gL) * sin(θ) * sqrt(1 - cos(θ))

**Simplifying the Induced Potential Difference Formula:**
- Using trigonometric identities, we can simplify the expression further:
sin(θ) * sqrt(1 - cos(θ)) = sin(θ) * sqrt(sin^2(θ)) = sin^2(θ)
Therefore, the potential difference formula becomes:
ΔV = B * L * sqrt(2gL) * sin^2(θ)

**Conclusion:**
The maximum potential difference induced across the simple pendulum with a bob mass m and conducting wire length L swinging through an angle 2θ under the Earth's magnetic field component B can be calculated using the formula ΔV = B * L * sqrt(2gL) * sin^2(θ). This potential difference is a