Problem 38 - University Physics with Modern Physics

Question

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?

Step-by-Step Solution

Answer

At 0.500 cm: \(1.02 \times 10^{-3}\) V/m; at 1.00 cm: \(2.04 \times 10^{-3}\) V/m.

Step 1: Understand the Problem

We have a solenoid with 900 turns/m and radius 2.50 cm. The current increases at 36.0 A/s. We need to find the induced electric field at two different radial distances (0.500 cm and 1.00 cm) from the solenoid axis.

Step 2: Recall Faraday's Law for Induced Electric Field

The induced electric field due to a changing magnetic field inside a solenoid can be described using Faraday's Law as: \[ E(2\pi r) = \mu_0 n \frac{di}{dt} \pi r^2 \]where \(E\) is the induced electric field, \(r\) is the distance from the center, \(\mu_0\) is the permeability of free space, \(n\) is the number of turns per unit length, and \(\frac{di}{dt}\) is the rate of change of current.

Step 3: Calculations for the Point 0.500 cm from Axis

Substitute the given values into the equation:\[ \frac{di}{dt} = 36.0 \text{ A/s}, \quad n = 900 \text{ turns/m}, \quad r = 0.005 \text{ m} \]The equation becomes:\[ E (2\pi \cdot 0.005) = (4\pi \times 10^{-7}) \times 900 \times 36 \times \pi \times (0.005)^2 \]Solve for \(E\):\[ E = \frac{(4\pi \times 10^{-7}) \times 900 \times 36 \times \pi \times (0.005)^2}{2\pi \times 0.005} \approx 1.02 \times 10^{-3} \text{ V/m} \]

Step 4: Calculations for the Point 1.00 cm from Axis

Use the same process with \(r = 0.010 \text{ m}\):\[ E (2\pi \cdot 0.010) = (4\pi \times 10^{-7}) \times 900 \times 36 \times \pi \times (0.010)^2 \]Solve for \(E\):\[ E = \frac{(4\pi \times 10^{-7}) \times 900 \times 36 \times \pi \times (0.010)^2}{2\pi \times 0.010} \approx 2.04 \times 10^{-3} \text{ V/m} \]

Step 5: Conclusion

The magnitude of the induced electric field at 0.500 cm is approximately \(1.02 \times 10^{-3} \text{ V/m}\), and at 1.00 cm, it is approximately \(2.04 \times 10^{-3} \text{ V/m}\).

Key Concepts

Induced Electric Field

The concept of an induced electric field emerges from the interaction between changing magnetic fields and electrical conductors. According to Faraday's Law of electromagnetic induction, when the magnetic field around a conductor changes, an electric field is induced. This electric field, in turn, can drive a current around a closed loop or circuit. The strength of this electric field depends on several factors:

• The rate at which the magnetic field changes.
• The area enclosed by the loop where the field is applied.

In the context of a solenoid, the induced electric field forms around the coil when the current inside the solenoid changes. This results in a magnetic field that also changes, thus creating a varying electric field along the solenoid's length. Understanding this field is essential for comprehending how electrical transformers and generators function in real-life applications.

Solenoid

A solenoid is a coil of wire designed to produce a magnetic field when an electric current passes through it. In physics, solenoids are often idealized as long coils where the length is much greater than the diameter. This allows for a relatively uniform magnetic field across most of the interior of the coil and a negligible field outside it.Here's what makes solenoids interesting:

• The magnetic field inside a solenoid is strong and nearly uniform, making it useful for various applications like electromagnets and inductors.
• The field strength (B) inside the solenoid can be calculated using the formula: B = \(\ \mu_0 n i \) where \(\ \mu_0 \) (permeability of free space), \(\ n \) is the number of turns per unit length, and \(\ i \) is the current.

The interior of a solenoid is where all the action happens, especially in exercises focused on calculating induced electric fields due to changing currents. The properties of solenoids make them fundamental in studying magnetism and electromagnetism.

Rate of Change of Current

The rate of change of current is a critical factor in determining the magnitude of an induced electric field. In a solenoid setup, it refers to how quickly the current flowing through the coils is changing over time. This change is essential because a changing current results in a varying magnetic field. Why is it important?

• The induced electric field strength is directly proportional to the rate at which the current changes. Simply put, faster changes in current magnify the resultant electric field.
• This concept is especially significant when designing devices like inductors, which rely on changing currents to function effectively.

Calculations involving the rate of change of current require the use of derivatives, as it aims to quantify the instantaneous rate of change. In the given exercise, a constant rate signifies a uniform magnetic alteration over time, producing a predictable induced electric field.

Magnetic Field

A magnetic field is a vector field that describes the magnetic influence of electric currents and magnetic materials. It's present around magnetic materials and moving charges and is fundamental to the workings of electric motors, generators, and magnetic storage systems. In a solenoid, the generated magnetic field lines form closed loops and are most concentrated inside the coil. Key characteristics of magnetic fields:

• They have both magnitude and direction; the direction of the magnetic field is the direction in which the north end of a compass needle points.
• In a solenoid, the magnetic field is concentrated and strong inside the coil while almost negligible outside.
• The strength of the field within a solenoid can be influenced by the number of turns in the coil, the current passing through, and the presence of a core material.

In the context of a changing current in a solenoid, understanding the behavior of magnetic fields helps in calculating the induced electric fields. These fields and their interactions are central to electromagnetic theory, offering insights into technological and natural phenomena.