The magnetic field inside a 5.0-cm-diameter solenoid is 2.0 T and decreasing at 4.0 T/s. What is the electric field strength inside the solenoid at a point (a) on the axis and (b) 2.0 cm from the axis?

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Understand the problem: The solenoid has a magnetic field that is decreasing over time, which induces an electric field according to Faraday's Law of Induction. We need to calculate the electric field strength at two points: (a) on the axis of the solenoid and (b) at a distance of 2.0 cm from the axis.

Apply Faraday's Law of Induction: The induced electric field is related to the rate of change of the magnetic flux. The magnetic flux through a loop of radius \( r \) is \( \Phi_B = B \cdot A = B \cdot \pi r^2 \), where \( B \) is the magnetic field and \( A \) is the area of the loop.

Differentiate the magnetic flux with respect to time: The rate of change of flux is \( \frac{d\Phi_B}{dt} = \pi r^2 \cdot \frac{dB}{dt} \), where \( \frac{dB}{dt} \) is the rate of change of the magnetic field (given as \( -4.0 \ \text{T/s} \)).

Relate the induced electric field to the rate of change of flux: The magnitude of the induced electric field at a distance \( r \) from the axis is given by \( E = \frac{1}{2\pi r} \cdot \left| \frac{d\Phi_B}{dt} \right| = \frac{1}{2} \cdot r \cdot \left| \frac{dB}{dt} \right| \). Substitute \( r = 0 \ \text{m} \) for part (a) and \( r = 0.02 \ \text{m} \) for part (b).

Substitute the known values: For part (a), since \( r = 0 \), the electric field on the axis is zero because the loop area is zero. For part (b), substitute \( r = 0.02 \ \text{m} \) and \( \frac{dB}{dt} = -4.0 \ \text{T/s} \) into the formula \( E = \frac{1}{2} \cdot r \cdot \left| \frac{dB}{dt} \right| \) to calculate the electric field strength.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Magnetic Field

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. In this context, the magnetic field inside the solenoid is given as 2.0 T (Tesla), which indicates the strength of the magnetic field. The behavior of the magnetic field is crucial for understanding how it interacts with electric fields and charges.

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 rate of change of the magnetic field is essential for calculating the induced electric field. In this case, the magnetic field is decreasing at a rate of 4.0 T/s, which will generate an electric field inside the solenoid.

Electric Field

An electric field is a region around a charged particle where other charged particles experience a force. The strength of the electric field induced by the changing magnetic field can be calculated using the relationship defined by Faraday's Law. The electric field strength varies with distance from the axis of the solenoid, which is important for determining the values at different points inside the solenoid.

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Textbook Question

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