Consider a short bar magnet forming a magnetic dipole enclosed by an imaginary co-axial cylindrical surface with circular base area. Magnet is at the middle of cylinder. If magnetic flux through one of the circular base is phi _(0) then the magnetic flux through the other circular base will be:

Options

A. `phi _(0)`
B. `gt phi _(0)`
C. `-phi _(0)`
D. `lt phi _(0)`

Explanation

To solve the problem, we need to analyze the situation involving a short bar magnet placed inside a cylindrical surface. The magnet creates a magnetic field, and we need to determine the magnetic flux through the two circular bases of the cylinder.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a short bar magnet placed in the center of a cylindrical surface with two circular bases (let's call them Base A and Base B).
- The bar magnet has a North Pole and a South Pole. Magnetic field lines emerge from the North Pole and enter the South Pole.

2. **Magnetic Flux Definition**:
- Magnetic flux (Φ) through a surface is defined as the product of the magnetic field (B) and the area (A) of the surface, considering the angle (θ) between the magnetic field and the normal to the surface:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]

3. **Flux through Base A**:
- Let’s denote the magnetic flux through Base A as Φ_A. According to the problem, this is given as Φ_0.

4. **Flux through Base B**:
- The magnetic field lines that enter Base A (the base facing the North Pole) will exit through Base B (the base facing the South Pole).
- Due to the conservation of magnetic field lines, the number of lines entering Base A is equal to the number of lines exiting Base B.

5. **Applying Gauss's Law for Magnetism**:
- According to Gauss's law for magnetism, the total magnetic flux through a closed surface is zero:
\[ \Phi_A + \Phi_B + \Phi_C = 0 \]
- Here, Φ_C is the flux through the cylindrical side surface, which is zero because no magnetic field lines pass through the curved surface of the cylinder.

6. **Setting Up the Equation**:
- Since Φ_C = 0, we can simplify the equation to:
\[ \Phi_A + \Phi_B = 0 \]
- This implies:
\[ \Phi_B = -\Phi_A \]

7. **Substituting the Known Value**:
- We know that Φ_A = Φ_0, therefore:
\[ \Phi_B = -\Phi_0 \]

8. **Conclusion**:
- The magnetic flux through the other circular base (Base B) is:
\[ \Phi_B = -\Phi_0 \]

### Final Answer:
The magnetic flux through the other circular base will be \(-\Phi_0\).

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