The magnetic field within a long, straight solenoid with a circular cross section and radius R is increasing at a rate of dB/dt. (e) What is the magnitude of the induced emf in a circular turn of radius R/2 that has its center on the solenoid axis?

Verified step by step guidance

1. Identify the relevant concepts: The problem involves the concept of electromagnetic induction, specifically Faraday's Law of Electromagnetic Induction, which states that the induced electromotive force (emf) in any closed circuit is equal to the negative rate of change of the magnetic flux through the circuit.
2. Determine the magnetic flux through the circular turn: The magnetic flux, \( \Phi \), through the circular turn of radius \( R/2 \) is given by the product of the magnetic field \( B \) and the area \( A \) of the circle. The area can be calculated using the formula \( A = \pi (R/2)^2 \).
3. Apply Faraday's Law: According to Faraday's Law, the magnitude of the induced emf \( \mathcal{E} \) is given by \( \mathcal{E} = |\frac{d\Phi}{dt}| \). Since the magnetic field is changing at a rate of \( \frac{dB}{dt} \), and the area of the circle is constant, the rate of change of the magnetic flux can be expressed as \( \frac{d\Phi}{dt} = A \frac{dB}{dt} \).
4. Substitute the expressions: Substitute the expression for the area \( A \) into the formula for \( \frac{d\Phi}{dt} \) to find the rate of change of magnetic flux through the circular turn.
5. Calculate the induced emf: Finally, use the expression for \( \frac{d\Phi}{dt} \) to find the magnitude of the induced emf using Faraday's Law. This will give you the final expression for the induced emf in terms of \( R \), \( \pi \), and \( \frac{dB}{dt} \).

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a closed loop induces an electromotive force (emf) in that loop. The induced emf is proportional to the rate of change of the magnetic flux, which is the product of the magnetic field strength and the area through which it passes. This principle is fundamental in understanding how changing magnetic fields can generate electric currents.

Magnetic Flux

Magnetic flux is a measure of the quantity of magnetism, taking into account the strength and the extent of a magnetic field. It is defined as the product of the magnetic field (B) and the area (A) perpendicular to the field through which it passes, expressed as Φ = B · A. In the context of the solenoid, the flux changes as the magnetic field strength increases, which is crucial for calculating the induced emf.

Induced EMF in a Loop

The induced emf in a circular loop placed in a varying magnetic field can be calculated using the formula ε = -dΦ/dt, where ε is the induced emf and dΦ/dt is the rate of change of magnetic flux through the loop. For a loop of radius R/2 centered on the axis of a solenoid, the area and the magnetic field strength at that radius must be considered to determine the magnitude of the induced emf accurately.