A solenoid is designed to produce a magnetic field of 0.0270 T at its center. It has radius 1.40 cm and length 40.0 cm, and the wire can carry a maximum current of 12.0 A. (a) What minimum number of turns per unit length must the solenoid have?

Verified step by step guidance

1. Identify the formula for the magnetic field at the center of a solenoid, which is given by B = \( \mu_0 \cdot n \cdot I \), where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, T \cdot m/A \)), \( n \) is the number of turns per unit length, and \( I \) is the current.
2. Rearrange the formula to solve for \( n \), the number of turns per unit length. This gives \( n = \frac{B}{\mu_0 \cdot I} \).
3. Substitute the values for \( B \) (0.0270 T) and \( I \) (12.0 A) into the rearranged formula.
4. Calculate \( n \) using the substituted values and the constant \( \mu_0 \).
5. Ensure that the calculated value of \( n \) meets the condition of being the minimum number of turns per unit length required to produce the specified magnetic field.

Key Concepts

Magnetic Field in a Solenoid

The magnetic field inside a long solenoid is given by 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. This relationship shows how the magnetic field strength is directly proportional to both the current flowing through the wire and the density of the turns.

Turns per Unit Length

Turns per unit length (n) refers to the number of loops of wire in a solenoid per unit of length. It is a crucial factor in determining the strength of the magnetic field produced by the solenoid. To find n, one can rearrange the magnetic field formula to n = B / (μ₀I), allowing for the calculation of the necessary turns to achieve a desired magnetic field.

Permeability of Free Space

The permeability of free space (μ₀) is a physical constant that describes how a magnetic field interacts with a vacuum. Its value is approximately 4π × 10⁻⁷ T·m/A. This constant is essential in the calculation of magnetic fields in solenoids and other electromagnetic devices, as it influences the relationship between magnetic field strength, current, and turns per unit length.