An Infinite current carrying wire having current `I_0` is placed coaxially inside a hollow conducting infinite cylinder of radius R and having current I flowing along the length of the cylinder in the same direction as `I_0` what is the increment in magnetic pressure due to placing of the wire?

Options

A. `(mu_0l_0l)/(2pi^2R^2)`
B. `(mu_0l_0l)/(2piR^2)`
C. `(mu_0I_0I)/(4pi^2R^2)`
D. `(mu_0l_0l)/(4piR^2)`

Answer

To solve the problem of determining the increment in magnetic pressure due to the placement of an infinite current-carrying wire inside a hollow conducting infinite cylinder, we can follow these steps:

### Step 1: Understand the Magnetic Field Due to the Infinite Wire
The magnetic field \( B \) at a distance \( r \) from an infinite straight wire carrying a current \( I_0 \) is given by Ampère's law:
\[ B = \frac{\mu_0 I_0}{2 \pi r} \]
where \( \mu_0 \) is the permeability of free space.

### Step 2: Determine the Magnetic Field Inside the Cylinder
Since the hollow conducting cylinder carries a current \( I \) in the same direction as \( I_0 \), the magnetic field inside the conducting cylinder (at radius \( r < R \)) will be the superposition of the fields due to both the wire and the cylinder. The magnetic field inside the cylinder is:
\[ B_{\text{cylinder}} = \frac{\mu_0 I}{2 \pi r} \]

### Step 3: Calculate the Total Magnetic Field Inside the Cylinder
The total magnetic field \( B_{\text{total}} \) at a distance \( r \) from the center (where \( r < R \)) is:
\[ B_{\text{total}} = B + B_{\text{cylinder}} = \frac{\mu_0 I_0}{2 \pi r} + \frac{\mu_0 I}{2 \pi r} = \frac{\mu_0 (I_0 + I)}{2 \pi r} \]

### Step 4: Calculate the Magnetic Pressure
The magnetic pressure \( P_B \) is given by the formula:
\[ P_B = \frac{B^2}{2 \mu_0} \]
Thus, the magnetic pressure due to the total magnetic field is:
\[ P_B = \frac{1}{2 \mu_0} \left( \frac{\mu_0 (I_0 + I)}{2 \pi r} \right)^2 \]
\[ = \frac{\mu_0 (I_0 + I)^2}{8 \pi^2 r^2} \]

### Step 5: Calculate the Increment in Magnetic Pressure
The increment in magnetic pressure \( \Delta P_B \) due to the placement of the wire is the difference between the magnetic pressure with and without the wire.

1. **Magnetic pressure without the wire (only the cylinder)**:
\[ P_{B,\text{cylinder}} = \frac{1}{2 \mu_0} \left( \frac{\mu_0 I}{2 \pi r} \right)^2 = \frac{\mu_0 I^2}{8 \pi^2 r^2} \]

2. **Magnetic pressure with the wire**:
\[ P_{B,\text{total}} = \frac{\mu_0 (I_0 + I)^2}{8 \pi^2 r^2} \]

Thus, the increment in magnetic pressure is:
\[ \Delta P_B = P_{B,\text{total}} - P_{B,\text{cylinder}} = \frac{\mu_0 (I_0 + I)^2}{8 \pi^2 r^2} - \frac{\mu_0 I^2}{8 \pi^2 r^2} \]
\[ = \frac{\mu_0}{8 \pi^2 r^2} \left( (I_0 + I)^2 - I^2 \right) \]
\[ = \frac{\mu_0}{8 \pi^2 r^2} \left( I_0^2 + 2I_0I \right) \]

### Final Result
The increment in magnetic pressure due to the placement of the wire is:
\[ \Delta P_B = \frac{\mu_0 I_0^2 + 2 \mu_0 I_0 I}{8 \pi^2 r^2} \]