Each turn of a solenoid is a current loop with a magnetic dipole moment. Consider a 200-turn cylindrical solenoid that has an interior volume of 40 cm³ and for which each turn is a magnetic dipole moment with magnitude 8.0 x 10^-4 A m². What is the magnetic field strength inside the solenoid?

Verified step by step guidance

1. Step 1: Understand the relationship between the magnetic field inside a solenoid and the magnetic dipole moment. The magnetic field strength inside a solenoid is proportional to the number of turns per unit length and the current flowing through the solenoid.
2. Step 2: Calculate the number of turns per unit length (n). Use the formula: n = N / L, where N is the total number of turns (200 turns) and L is the length of the solenoid. To find L, use the volume of the solenoid (40 cm³) and assume the cross-sectional area is uniform.
3. Step 3: Relate the magnetic dipole moment per turn to the current. The magnetic dipole moment (μ) for each turn is given as 8.0 x 10⁻⁴ A·m². Use the formula μ = I × A, where I is the current and A is the cross-sectional area of the solenoid, to find the current.
4. Step 4: Use the formula for the magnetic field inside a solenoid: B = μ₀ × n × I, where μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A), n is the number of turns per unit length, and I is the current. Substitute the values calculated in previous steps.
5. Step 5: Simplify the expression to find the magnetic field strength inside the solenoid. Ensure all units are consistent (e.g., convert cm³ to m³ for volume) before substituting values.

Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that represents the strength and orientation of a magnetic source. For a current loop, it is defined as the product of the current flowing through the loop and the area of the loop. In this case, each turn of the solenoid contributes to the overall magnetic dipole moment, which influences the magnetic field generated by the solenoid.

Solenoid

A solenoid is a long coil of wire that generates a magnetic field when an electric current passes through it. The magnetic field inside a solenoid is uniform and can be calculated using the formula B = μ₀(nI), where B is the magnetic field strength, μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current. The properties of the solenoid, such as the number of turns and its dimensions, directly affect the strength of the magnetic field.

Magnetic Field Strength

Magnetic field strength, often denoted as B, is a measure of the magnetic force experienced by a unit magnetic pole in a magnetic field. It is expressed in teslas (T) and is influenced by factors such as the current flowing through the solenoid and the number of turns. In a solenoid, the magnetic field strength can be derived from the magnetic dipole moments of the individual turns, allowing for the calculation of the total magnetic field within the solenoid.