FIGURE EX30.19 shows the current as a function of time through a 20-cm-long, 4.0-cm-diameter solenoid with 400 turns. Draw a graph of the induced electric field strength as a function of time at a point 1.0 cm from the axis of the solenoid.

Verified step by step guidance

1. Step 1: Understand the problem. The solenoid has a changing current, which induces an electric field. The graph provided shows the current (I) as a function of time (t). The goal is to calculate the induced electric field strength at a point 1.0 cm from the axis of the solenoid and plot it as a function of time.
2. Step 2: Recall Faraday's Law of Induction. The induced electric field is related to the rate of change of magnetic flux through the solenoid. The magnetic flux (Φ_B) is given by Φ_B = B × A, where B is the magnetic field inside the solenoid and A is the cross-sectional area of the solenoid. The magnetic field inside the solenoid is B = μ₀ × n × I, where μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current.
3. Step 3: Calculate the rate of change of magnetic flux. The induced electric field strength (E) at a distance r from the axis is given by E = (1 / (2πr)) × (dΦ_B / dt). To find dΦ_B / dt, differentiate Φ_B with respect to time, using the expression for B and A. Note that A = π × (radius of solenoid)^2.
4. Step 4: Analyze the graph of current vs. time. The graph shows linear segments where the current increases, remains constant, and decreases. For each segment, calculate dI/dt (rate of change of current) and use it to find dΦ_B / dt. Substitute these values into the formula for E to determine the induced electric field strength at r = 1.0 cm.
5. Step 5: Plot the graph of E vs. t. Using the calculated values of E for each time interval, draw the graph. The shape of the graph will correspond to the changes in dI/dt: increasing during the rising current, zero during constant current, and decreasing during the falling current.

Key Concepts

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field within a closed loop induces an electromotive force (EMF) in the loop. This phenomenon is described by Faraday's Law, which states that the induced EMF is proportional to the rate of change of magnetic flux through the loop. In the context of the solenoid, as the current changes over time, it creates a varying magnetic field that induces an electric field in the surrounding space.

Induced Electric Field

The induced electric field is generated in response to a changing magnetic field, as described by Faraday's Law of Induction. The strength of this electric field can be calculated using the formula E = -dΦ/dt, where E is the electric field strength and Φ is the magnetic flux. The induced electric field circulates around the axis of the solenoid and its strength varies with the distance from the solenoid and the rate of change of current.

Solenoid Characteristics

A solenoid is a coil of wire designed to create a uniform magnetic field when an electric current passes through it. The characteristics of a solenoid, such as its length, diameter, and number of turns, influence the strength of the magnetic field it produces. In this case, the solenoid has 400 turns and a specific length and diameter, which will affect the induced electric field at a point 1.0 cm from its axis as the current changes over time.