The magnetic field within a long, straight solenoid with a circular cross section and radius R is increasing at a rate of dB/dt. What is the magnitude of the induced emf if the radius in part (d) is 2R?

Verified step by step guidance

1. Understand that the problem involves electromagnetic induction, specifically Faraday's law of induction, which states that a changing magnetic field within a closed loop induces an electromotive force (emf) in the loop.
2. Recall Faraday's law of induction, which can be expressed as: $\epsilon =-\frac{d\Phi }{dt}$, where $\epsilon$ is the induced emf and $\Phi$ is the magnetic flux.
3. Calculate the magnetic flux $\Phi$ through a loop of radius 2R. The magnetic flux is given by: $\Phi =BA$, where $B$ is the magnetic field and $A$ is the area of the loop. For a loop of radius 2R, the area is $A=\pi (2R{)}^2$.
4. Substitute the expression for the area into the magnetic flux equation: $\Phi =B\pi (2R{)}^2$. Differentiate this expression with respect to time to find the rate of change of magnetic flux: $\frac{d\Phi }{dt}=\pi (2R{)}^2\frac{dB}{dt}$.
5. Apply Faraday's law to find the induced emf: $\epsilon =-\pi (2R{)}^2\frac{dB}{dt}$. The negative sign indicates the direction of the induced emf according to Lenz's law, which opposes the change in magnetic flux.

Key Concepts

Faraday's Law of Induction

Faraday's Law states that a change in magnetic flux through a loop induces an electromotive force (emf) in the loop. The induced emf is proportional to the rate of change of the magnetic flux. In this scenario, the changing magnetic field within the solenoid induces an emf in the surrounding area.

Magnetic Flux

Magnetic flux is the product of the magnetic field and the area it penetrates, perpendicular to the field lines. It is a measure of the quantity of magnetism, considering the strength and extent of a magnetic field. Understanding how flux changes with the solenoid's radius is crucial for calculating the induced emf.

Solenoid Magnetic Field

A solenoid is a coil of wire that generates a uniform magnetic field when an electric current passes through it. The magnetic field inside a long, straight solenoid is directly proportional to the current and the number of turns per unit length. The field's increase rate, dB/dt, is essential for determining the induced emf in the surrounding area.