A tightly- wound solenoid of radius a and length l has n turns per unit length. It carries an electric current i. Consider a length dx of the solenoid at a distance x from one end. This contains n dx turns and may be approximated as a circular current I n dx. (a) Write the magnetic field at the center of the solenoid due to this circular current. Integrate this expression under proper limits to find the magnetic field at the center of the solenoid. (b) Verify that if `lgtgta`, the field tends to `(B=(mu_0ni)` and if `agtgt1`, the field tends to` B= ((mu_0)nil/2a)` . Interpret these results.

Explanation

To solve the problem step by step, we will break it down into two parts as given in the question.

### Part (a): Magnetic Field at the Center of the Solenoid

1. **Understanding the Setup**:
- We have a tightly wound solenoid of radius \( a \) and length \( l \) with \( n \) turns per unit length. It carries a current \( i \).
- We consider a small section of the solenoid of length \( dx \) at a distance \( x \) from one end. This section contains \( n \, dx \) turns.

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