A constant current I is maintained in a solenoid. Which of lthe following quantities will increase if an iron rod is inserted in the solenoid along its axis?

Options

A. magnetic field at the centre
B. magnetic flux linked with the solenoid
C. self inductance of the solenoid
D. rate of Joule heating.

Answer

To solve the problem, we need to analyze the effects of inserting an iron rod into a solenoid with a constant current flowing through it. Let's break down the solution step by step:

### Step 1: Understanding the Solenoid
A solenoid is a coil of wire that generates a magnetic field when an electric current passes through it. The magnetic field (B) inside a long solenoid is given by the formula:

\[ B = \mu_0 n I \]

where:
- \( \mu_0 \) is the permeability of free space,
- \( n \) is the number of turns per unit length of the solenoid,
- \( I \) is the current flowing through the solenoid.

### Step 2: Inserting the Iron Rod
When an iron rod is inserted into the solenoid, the relative permeability (\( \mu_r \)) of the iron is greater than 1. The magnetic field in the solenoid with the iron rod can be expressed as:

\[ B = \mu_0 \mu_r n I \]

Since \( \mu_r > 1 \), it follows that the magnetic field \( B \) will increase when the iron rod is inserted.

### Step 3: Magnetic Flux
The magnetic flux (\( \Phi \)) linked with the solenoid is given by the product of the magnetic field and the cross-sectional area (A) of the solenoid:

\[ \Phi = B \cdot A \]

Since we established that \( B \) increases due to the insertion of the iron rod, the magnetic flux \( \Phi \) will also increase.

### Step 4: Self-Inductance
The self-inductance (L) of the solenoid is related to the magnetic flux and the current by the formula:

\[ L = \frac{\Phi}{I} \]

As the magnetic flux \( \Phi \) increases (while the current \( I \) remains constant), the self-inductance \( L \) will also increase.

### Conclusion
In summary, when an iron rod is inserted into the solenoid:
- The magnetic field \( B \) increases.
- The magnetic flux \( \Phi \) linked with the solenoid increases.
- The self-inductance \( L \) of the solenoid increases.

Thus, the quantities that will increase are:
- Magnetic field (B)
- Magnetic flux (Φ)
- Self-inductance (L)

### Final Answer
The correct options are A, B, and C.

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