Find the force per unit area on the surface of a long hollow cylinder carrying uniform current `I` and radius of the cylinder is R.

Answer

To find the force per unit area on the surface of a long hollow cylinder carrying a uniform current \( I \) and having a radius \( R \), we can follow these steps:

### Step 1: Understand the Magnetic Field Around the Cylinder
For a long straight current-carrying conductor, the magnetic field \( B \) at a distance \( r \) from the axis of the cylinder is given by:
\[ B = \frac{\mu_0 I}{2\pi r} \]
where \( \mu_0 \) is the permeability of free space.

### Step 2: Determine the Magnetic Field at the Surface
At the surface of the hollow cylinder (where \( r = R \)), the magnetic field becomes:
\[ B = \frac{\mu_0 I}{2\pi R} \]

### Step 3: Calculate the Force on a Small Element of the Cylinder
The force \( dF \) on a small current element \( dI \) of the cylinder in the magnetic field \( B \) can be expressed using the formula:
\[ dF = I \, dL \times B \]
where \( dL \) is the length of the current element. Since the current is uniformly distributed, we can consider a small angle \( d\theta \) at the circumference.

### Step 4: Express the Current Element
The current \( dI \) through a small angle \( d\theta \) at the surface of the cylinder can be expressed as:
\[ dI = \frac{I}{2\pi} \cdot d\theta \]
This is derived from the fact that the total current \( I \) is distributed uniformly around the circumference \( 2\pi R \).

### Step 5: Calculate the Force on the Small Element
The length \( dL \) can be considered as \( R \, d\theta \) (the arc length). Therefore, the force on this small element becomes:
\[ dF = dI \cdot B \cdot R \, d\theta \]
Substituting \( dI \) and \( B \):
\[ dF = \left(\frac{I}{2\pi} d\theta\right) \cdot \left(\frac{\mu_0 I}{2\pi R}\right) \cdot R \, d\theta \]
Simplifying this gives:
\[ dF = \frac{\mu_0 I^2}{4\pi^2} d\theta \]

### Step 6: Calculate the Total Force
To find the total force \( F \) on the entire surface, integrate \( dF \) over the angle from \( 0 \) to \( 2\pi \):
\[ F = \int_0^{2\pi} dF = \int_0^{2\pi} \frac{\mu_0 I^2}{4\pi^2} d\theta = \frac{\mu_0 I^2}{4\pi^2} \cdot 2\pi = \frac{\mu_0 I^2}{2\pi} \]

### Step 7: Calculate the Area of the Cylinder
The area \( A \) of the cylindrical surface is given by:
\[ A = 2\pi R L \]
where \( L \) is the length of the cylinder.

### Step 8: Calculate the Force per Unit Area
The force per unit area \( \frac{F}{A} \) is then:
\[ \frac{F}{A} = \frac{\frac{\mu_0 I^2}{2\pi}}{2\pi R L} = \frac{\mu_0 I^2}{4\pi^2 R L} \]

### Conclusion
Thus, the force per unit area on the surface of the hollow cylinder is:
\[ \frac{F}{A} = \frac{\mu_0 I^2}{4\pi^2 R} \]