A closely wound solenoid 80 cm long has 5 layers of windings of 400

turns each. The diameter of the solenoid is 1.8 cm. If the current

carried is 8.0 A, estimate the magnitude of

B inside the solenoid

near its centre.

To find the magnetic field strength (B) inside a solenoid, we can use the formula derived from Ampère's law. The magnetic field inside a long solenoid is given by the equation:

Magnetic Field in a Solenoid

The formula for the magnetic field inside a solenoid is:

B = μ₀ * n * I

Where:

• B = magnetic field strength (in teslas, T)
• μ₀ = permeability of free space (4π × 10^-7 T·m/A)
• n = number of turns per unit length (in turns/m)
• I = current through the solenoid (in amperes, A)

Step-by-Step Calculation

Let’s break down the calculation into manageable steps:

1. Calculate the Total Number of Turns

The solenoid has 5 layers of windings, with each layer containing 400 turns. Therefore, the total number of turns (N) is:

N = 5 layers × 400 turns/layer = 2000 turns

2. Determine the Length of the Solenoid

The length of the solenoid is given as 80 cm, which we need to convert to meters:

Length (L) = 80 cm = 0.8 m

3. Calculate the Number of Turns per Unit Length (n)

Now, we can find the number of turns per unit length:

n = N / L = 2000 turns / 0.8 m = 2500 turns/m

4. Plug Values into the Magnetic Field Formula

Now that we have n, we can substitute it along with the current (I = 8.0 A) and the permeability of free space (μ₀ = 4π × 10^-7 T·m/A) into the formula:

B = (4π × 10^-7 T·m/A) × (2500 turns/m) × (8.0 A)

5. Perform the Calculation

Calculating this gives:

B = (4π × 10^-7) × 2500 × 8.0

First, calculate the constant:

4π × 10^-7 ≈ 1.2566 × 10^-6 T·m/A

Now, substituting this value:

B ≈ (1.2566 × 10^-6) × 2500 × 8.0

B ≈ 2.51 × 10^-3 T

Final Result

The magnitude of the magnetic field (B) inside the solenoid near its center is approximately 2.51 mT (milliteslas).

This calculation illustrates how the magnetic field strength inside a solenoid is directly proportional to the number of turns per unit length and the current flowing through it. The more turns and the higher the current, the stronger the magnetic field produced. This principle is fundamental in electromagnetism and has numerous applications in technology, such as in inductors, transformers, and magnetic resonance imaging (MRI).