A long, thin solenoid has 900 turns per meter and radius 2.50 cm. The current in the solenoid is increasing at a uniform rate of 36.0 A/s. What is the magnitude of the induced electric field at a point near the center of the solenoid and (a) 0.500 cm from the axis of the solenoid; (b) 1.00 cm from the axis of the solenoid?

Verified step by step guidance

1. Understand that the problem involves a solenoid with a changing current, which induces an electric field. The solenoid has 900 turns per meter, a radius of 2.50 cm, and the current is increasing at 36.0 A/s.
2. Recall Faraday's law of electromagnetic induction, which states that a changing magnetic field within a closed loop induces an electromotive force (EMF) in the loop. The induced electric field (E) can be found using the formula: \( E = \frac{r}{2} \frac{dB}{dt} \), where \( r \) is the distance from the axis and \( \frac{dB}{dt} \) is the rate of change of the magnetic field.
3. Calculate the rate of change of the magnetic field \( \frac{dB}{dt} \) inside the solenoid using the formula: \( \frac{dB}{dt} = \mu_0 n \frac{dI}{dt} \), where \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \) T·m/A), \( n \) is the number of turns per unit length (900 turns/m), and \( \frac{dI}{dt} \) is the rate of change of current (36.0 A/s).
4. For part (a), substitute \( r = 0.500 \) cm (converted to meters) into the formula for the induced electric field: \( E = \frac{r}{2} \frac{dB}{dt} \). Calculate \( E \) using the previously found \( \frac{dB}{dt} \).
5. For part (b), repeat the process with \( r = 1.00 \) cm (converted to meters) to find the induced electric field at this distance from the axis of the solenoid.

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a changing magnetic field within a closed loop induces an electromotive force (EMF) in the loop. The induced EMF is proportional to the rate of change of the magnetic flux through the loop. In the context of a solenoid, the changing current alters the magnetic field, inducing an electric field in the surrounding space.

Magnetic Field Inside a Solenoid

The magnetic field inside a long, thin solenoid is uniform and parallel to the axis of the solenoid. It is given by B = μ₀nI, where μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current. This uniform field is crucial for calculating the change in magnetic flux, which is needed to determine the induced electric field.

Induced Electric Field in a Solenoid

The induced electric field in a solenoid due to a changing current can be calculated using the relationship E = (r/2) * (dB/dt), where r is the radial distance from the axis, and dB/dt is the rate of change of the magnetic field. This formula arises from integrating the induced EMF around a circular path concentric with the solenoid, highlighting how the field varies with distance from the axis.