Question

The number of turns in a solenoid is doubled, and its length is halved. How does its magnetic field change? a) it doubles b) it is halved c) it quadruples d) it remains unchanged

Short answer

Answer: The magnetic field quadruples.

Step-by-step solution

Identify the formula for the magnetic field of a solenoid

To determine how the magnetic field changes, we first need to know the formula for the magnetic field of a solenoid. The formula is given by: B = μ₀ * n * I where B is the magnetic field, μ₀ is the magnetic constant (also known as the permeability of free space), n is the number of turns per unit length, and I is the current flowing through the solenoid. We'll assume that the current I stays the same because the exercise doesn't mention it changing.

Analyze the changes in the number of turns and length of the solenoid

The exercise states that the number of turns in the solenoid is doubled, so we will multiply the number of turns (n) by 2: n_new = 2 * n It also says that the length of the solenoid is halved, so we will divide the length (L) by 2: L_new = L / 2

Calculate the new number of turns per unit length

Since the number of turns per unit length is given by n = N / L, we can express the new number of turns per unit length (n_new) with the new values of N and L: n_new = N_new / L_new n_new = (2 * N) / (L / 2) Simplifying the above expression, we get: n_new = 4 * (N / L) Comparing this to the original number of turns per unit length: n_new = 4 * n

Calculate the new magnetic field and compare to the original

Now that we have the new number of turns per unit length, we can calculate the new magnetic field (B_new) using the original formula: B_new = μ₀ * n_new * I Replacing n_new with the expression we found in Step 3: B_new = μ₀ * (4 * n) * I Comparing this to the original magnetic field: B_new = 4 * (μ₀ * n * I) We find that the new magnetic field is four times the original magnetic field: B_new = 4 * B Therefore, the correct answer is (c) the magnetic field quadruples.

Key concepts

Number of Turns in Solenoid

A solenoid is essentially a coil of wire, and the number of turns refers to how many loops or coils the wire makes. This is critical because each loop generates its own magnetic field. When the wire is coiled, these individual magnetic fields combine to create a stronger total magnetic field inside the solenoid.

• More turns mean more loops, hence a stronger magnetic field.
• Doubling the number of turns in a solenoid doubles the contribution each loop adds to the total magnetic field.

Thus, increasing the number of turns amplifies the overall magnetic field within a solenoid, especially when other factors, such as current, are kept constant.

Length of Solenoid

The length of a solenoid is another key factor that affects its magnetic field. The length determines the distribution of the turns within the solenoid, which in turn affects how concentrated the magnetic field is.

• By halving the length, you effectively concentrate the same number of turns in a smaller space.
• This concentration increases the number of turns per unit length.

The increase in turns per unit length when the solenoid's length is reduced results in a stronger magnetic field. So, even with the same overall number of turns, a shorter solenoid creates a more intense magnetic environment inside.

Magnetic Constant

The magnetic constant, often denoted as \( \mu_0 \), is an integral part of understanding solenoids. It represents the permeability of free space, which is a measure of how easily a magnetic field can form in a vacuum.

• The magnetic constant provides a baseline to calculate how a material will interact with magnetic fields.
• It remains unchanged regardless of changes to the solenoid's physical characteristics, like length or the number of turns.

When calculating the magnetic field inside a solenoid, the magnetic constant scales the effect of the current and turn density. It ensures that the calculated magnetic field is accurate and reflects the true strength of the solenoid's magnetic field in practical applications.