A sphere of radius R is placed near a long straight wire that carries a steady current l. The magnetic field generated by the current is B. The total magnetic flux passing through the sphere is

Options

A. `mu_(0)l`
B. `(mu_(0)l)/(4piR^(2))`
C. Zero
D. Need more information

Answer

To solve the problem of finding the total magnetic flux passing through a sphere placed near a long straight wire carrying a steady current, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**: We have a long straight wire carrying a current \( I \) and a sphere of radius \( R \) placed near this wire. The magnetic field \( B \) generated by the current in the wire will affect the sphere.

2. **Magnetic Field Due to the Wire**: The magnetic field \( B \) at a distance \( D \) from a long straight wire carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2 \pi D} \]
where \( \mu_0 \) is the permeability of free space.

3. **Magnetic Flux Definition**: The magnetic flux \( \Phi \) through a surface is defined as:
\[ \Phi = \int B \cdot dS \]
where \( dS \) is an infinitesimal area element of the surface through which the magnetic field lines pass.

4. **Applying Gauss's Law for Magnetism**: According to Gauss's law for magnetism, the net magnetic flux through any closed surface is zero. This is because magnetic field lines are continuous loops; they do not begin or end at any point (there are no magnetic monopoles). Thus, the number of magnetic field lines entering a closed surface equals the number of lines leaving it.

5. **Conclusion**: Since the sphere is a closed surface, the total magnetic flux \( \Phi \) through the sphere is:
\[ \Phi = 0 \]

### Final Answer:
The total magnetic flux passing through the sphere is \( \Phi = 0 \).

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