To find the magnetic field at point P on the axis at the midpoint of a solenoid, we can use the formula for the magnetic field inside a long solenoid. The formula is given by:
Magnetic Field Formula
The magnetic field \( B \) inside a solenoid is calculated using the equation:
B = \mu_0 \cdot n \cdot I
Where:
- B = magnetic field (in teslas)
- \(\mu_0\) = permeability of free space (given as \( 4 \pi \times 10^{-7} \, \text{H/m} \))
- n = number of turns per unit length (in turns/meter)
- I = current (in amperes)
Step-by-Step Calculation
Let’s break down the calculation into manageable steps:
Step 1: Calculate the Number of Turns per Unit Length (n)
The total number of turns in the solenoid is 1000, and the length of the solenoid is 0.4 m. Thus, we can find \( n \) as follows:
n = \frac{\text{Total Turns}}{\text{Length}} = \frac{1000}{0.4} = 2500 \, \text{turns/m}
Step 2: Substitute Values into the Magnetic Field Formula
Now that we have \( n \), we can substitute the values into the magnetic field formula:
B = \mu_0 \cdot n \cdot I
Substituting the known values:
- \(\mu_0 = 4 \pi \times 10^{-7} \, \text{H/m}\)
- n = 2500 turns/m
- I = 5.0 × 10^{-3} A
Thus, we have:
B = (4 \pi \times 10^{-7}) \cdot (2500) \cdot (5.0 \times 10^{-3})
Step 3: Calculate the Magnetic Field
Now, let’s perform the calculation step-by-step:
- First, calculate \( 4 \pi \times 10^{-7} \):
- Using \( \pi \approx 3.14 \), we get \( 4 \times 3.14 \times 10^{-7} \approx 1.256 \times 10^{-6} \, \text{H/m} \)
- Next, multiply by \( n \) and \( I \):
B = (1.256 \times 10^{-6}) \cdot (2500) \cdot (5.0 \times 10^{-3})
Calculating this gives:
- First, \( 1.256 \times 2500 = 3140 \times 10^{-6} \)
- Then, \( 3140 \times 5.0 \times 10^{-3} = 15.7 \times 10^{-3} \, \text{T} = 1.57 \times 10^{-2} \, \text{T} \)
Step 4: Convert to Weber/m²
Since \( 1 \, \text{T} = 1 \, \text{Wb/m}^2 \), we can express the magnetic field as:
B = 1.57 \times 10^{-2} \, \text{Wb/m}^2 = 15.7 \times 10^{-6} \, \text{Wb/m}^2
Final Answer
After calculating, we find that the magnetic field at point P on the axis at the midpoint of the solenoid is approximately:
15.7 × 10^{-6} \, \text{Wb/m}^2
However, this value does not match any of the provided options. It’s important to double-check the calculations and assumptions made during the process. If the values provided in the question are accurate, the magnetic field at the midpoint of the solenoid should be around \( 1.57 \times 10^{-5} \, \text{Wb/m}^2 \), which suggests that there might be a discrepancy in the options given. Please verify the parameters or the options provided.
