A hollow tube is carrying an electric current along its length distributed uniformly over its surface. The magnetic field

Options

A. increase linearly from the axis to the surface.
B. is constant inside the tube
C. is zero at the axis
D. is zero just outside the tude.

Answer

To solve the problem of determining the magnetic field inside and outside a hollow tube carrying an electric current uniformly distributed over its surface, we can follow these steps:

### Step 1: Understand the Configuration
We have a hollow tube with a uniform current distributed over its surface. The current flows along the length of the tube.

**Hint:** Visualize the hollow tube and the direction of the current flowing along its length.

### Step 2: Apply Ampere's Circuital Law
To find the magnetic field, we can use Ampere's Circuital Law, which states:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}} \]
where \(I_{\text{enc}}\) is the current enclosed by the Amperian loop.

**Hint:** Remember that the Amperian loop is a closed path around which we will calculate the magnetic field.

### Step 3: Magnetic Field Inside the Tube
At the axis of the tube, there is no current enclosed by the Amperian loop since the current is only on the surface. Therefore:
\[ I_{\text{enc}} = 0 \]
This leads to:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = 0 \]
Thus, the magnetic field \(B\) at the axis of the tube is:
\[ B = 0 \]

**Hint:** Consider that the magnetic field inside a hollow conductor with no enclosed current is zero.

### Step 4: Magnetic Field Outside the Tube
Now, consider a point outside the tube. The current is uniformly distributed over the surface of the tube. For a circular Amperian loop of radius \(r\) (where \(r\) is greater than the radius of the tube), the enclosed current \(I\) is equal to the total current flowing through the tube.

Using Ampere’s Law:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = B \cdot (2\pi r) = \mu_0 I \]
From this, we can solve for \(B\):
\[ B = \frac{\mu_0 I}{2\pi r} \]

**Hint:** Remember that as you move further away from the tube (increasing \(r\)), the magnetic field decreases with \(1/r\).

### Step 5: Summary of Magnetic Field Behavior
- **Inside the tube (at the axis):** \(B = 0\)
- **Outside the tube:** \(B = \frac{\mu_0 I}{2\pi r}\) which decreases with increasing distance \(r\).

### Conclusion
The magnetic field is zero at the axis of the tube and increases as you move towards the surface of the tube, becoming non-zero outside the tube.

**Final Answer:**
- The magnetic field is zero at the axis (option C is correct).
- The magnetic field is constant inside the tube (option B is correct).
- The magnetic field increases linearly from the axis to the surface (this is incorrect).
- The magnetic field is not zero just outside the tube (this is incorrect).

**Correct Options:** B and C.