Problem 5 - University Physics with Modern Physics

Question

Two toroidal solenoids are wound around the same form so that the magnetic field of one passes through the turns of the other. Solenoid 1 has 700 turns, and solenoid 2 has 400 turns. When the current in solenoid 1 is 6.52 A, the average flux through each turn of solenoid 2 is 0.0320 Wb. (a) What is the mutual inductance of the pair of solenoids? (b) When the current in solenoid 2 is 2.54 \(\mathrm{A}\) , what is the average flux through each turn of solenoid 1\(?\)

Step-by-Step Solution

Answer

(a) 0.00491 H, (b) 0.0125 Wb.

Step 1: Understanding Mutual Inductance

Mutual inductance, \( M \), between two solenoids can be found using the formula for the magnetic flux through solenoid 2 due to the current in solenoid 1. This formula is \( \Phi_{21} = M \cdot I_1 \), where \( \Phi_{21} \) is the flux through solenoid 2 and \( I_1 \) is the current in solenoid 1.

Step 2: Calculate the Mutual Inductance

We know that the flux \( \Phi_{21} = 0.0320 \) Wb and \( I_1 = 6.52 \) A. Using \( M = \frac{\Phi_{21}}{I_1} \), we plug in the values to get \( M = \frac{0.0320}{6.52} \approx 0.00491 \text{ H (henries)} \).

Step 3: Understanding Flux Through Solenoid 1

When the current flows through solenoid 2 instead of solenoid 1, we use a similar approach as before to understand how the same mutual inductance now affects solenoid 1.

Step 4: Calculate the Flux Through Each Turn of Solenoid 1

Using the mutual inductance \( M = 0.00491 \) H and the current \( I_2 = 2.54 \) A in solenoid 2, the flux through each turn of solenoid 1 can be found with \( \Phi_{12} = M \cdot I_2 \). Substitute the known values, \( \Phi_{12} = 0.00491 \times 2.54 \approx 0.0125 \) Wb.

Key Concepts

Magnetic Flux

Magnetic flux is a critical concept in understanding how magnetic fields interact with different materials, especially conductors like solenoids. It is often symbolized by the Greek letter \( \Phi \) and measures the total magnetic field passing through a given area. Think of magnetic flux as the number of magnetic field lines that pass through a surface.

Magnetic flux depends on the strength of the magnetic field, the area through which it passes, and the orientation relative to the field. When the field lines are perpendicular to the surface, the flux is maximized. Conversely, if the field lines are parallel to the surface, the flux is zero.

• Measured in Webers (Wb).
• Affected by changes in current and coil configuration.
• Vital for calculating mutual inductance.

Solenoid

A solenoid is a coil of wire that produces a magnetic field when an electric current flows through it. Solenoids are widely used in scientific and industrial applications due to their efficiency in creating uniform magnetic fields. These devices are key in various mechanisms, including relays, speakers, and even in forming the basis of electromagnets.

When discussing solenoids, we usually consider their:

• Turns of wire: More turns mean a stronger magnetic field.
• Current: Higher currents enhance the magnetic field strength.
• Core material: Materials like iron can markedly strengthen the field.

In the exercise, two solenoids are discussed, which interact through mutual inductance.

Magnetic Field

The magnetic field is an invisible force field around magnetized materials, influenced by and exerting force on charged particles and currents. Often represented by the symbol \( B \) and measured in Teslas (T), these fields can be uniform or vary over a specific distance.

Magnetic fields play a significant role in understanding how solenoids operate:

• Direction: Determined by the right-hand rule relative to current direction.
• Strength: Proportional to the current and number of turns in a solenoid.
• Interaction: Causes electromechanical forces, enabling devices like motors.

Information on magnetic fields is essential when calculating mutual inductance, as it directly affects the magnitude of the induced current and magnetic flux through solenoids.

Toroidal Solenoid

A toroidal solenoid is a specialized version of a standard solenoid, shaped like a doughnut or ring. Its design ensures that the magnetic field remains contained within the coil, minimizing external magnetic interference and energy losses.

Here’s how toroidal solenoids stand out:

• Efficiency: Toroidal shape reduces energy stray fields compared to linear solenoids.
• Field Configuration: Magnetic field forms a closed loop, primarily contained within the core.
• Appeal in Applications: Used in applications requiring minimal magnetic leakage, like transformers and inductors.

In the exercise, the toroidal solenoids are arranged so that the magnetic field of one passes through the turns of the other, illustrating mutual inductance firsthand.