Step 1: Understanding the Concept:
A solenoid is a long coil of wire wrapped in many close turns. When an electric current flows through it, it generates a magnetic field.
Inside a "long" or "ideal" solenoid, the magnetic field is remarkably uniform and directed along the axis of the solenoid.
Outside the solenoid, the field is practically zero.
The strength of this internal field depends solely on the number of turns per unit length and the current, assuming the core is air or vacuum.
This behavior is described by Ampere's Circuital Law.
Step 2: Key Formula or Approach:
The magnitude of the magnetic field (\(B\)) inside an ideal solenoid is given by the formula:
\[ B = \mu_{0} n I \]
Where:
\( \mu_{0} = 4\pi \times 10^{-7} \text{ T m A}^{-1} \) is the permeability of free space.
\( n \) is the number of turns per unit length (turns/meter).
\( I \) is the current in Amperes (A).
Step 3: Detailed Explanation:
Let's list the values provided:
1. Turn density (\(n\)): The question states "500 turns per meter".
So, \( n = 500 \text{ m}^{-1} \).
2. Current (\(I\)): The current is given as 5 A.
3. Permeability (\( \mu_{0} \)): \( 4\pi \times 10^{-7} \).
Now, perform the multiplication according to the formula:
\[ B = (4\pi \times 10^{-7}) \times 500 \times 5 \]
Group the numbers to make the calculation easier:
\[ B = 4\pi \times (500 \times 5) \times 10^{-7} \]
\[ B = 4\pi \times (2500) \times 10^{-7} \]
\[ B = 10000 \pi \times 10^{-7} \]
Express 10,000 as a power of 10: \( 10,000 = 10^{4} \).
\[ B = \pi \times 10^{4} \times 10^{-7} \]
Using the exponent rule \( 10^{a} \times 10^{b} = 10^{a+b} \):
\[ B = \pi \times 10^{4-7} \]
\[ B = \pi \times 10^{-3} \text{ T} \]
This uniform field exists deep inside the solenoid and its direction is determined by the Right-Hand Grip Rule.
Step 4: Final Answer:
The magnetic field intensity inside the solenoid is \( \pi \times 10^{-3} \) Tesla.