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 magnetic flux through the two circular bases of a cylindrical surface that encloses a short bar magnet placed at the center.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a cylindrical surface with a bar magnet placed at its center.
- The bar magnet has a North Pole and a South Pole.

**Hint**: Visualize the setup by sketching a cylinder with a bar magnet in the middle. Identify the North and South poles.

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

**Hint**: Remember that the angle θ is important when calculating flux; it affects how much of the magnetic field passes through the surface.

3. **Magnetic Field Lines**:
- The magnetic field lines emerge from the North Pole and terminate at the South Pole. Since the magnet is symmetrically placed in the cylinder, the field lines will be evenly distributed.

**Hint**: Think about how the magnetic field lines behave in a dipole configuration. They spread out from the North and converge at the South.

4. **Flux through the Circular Bases**:
- Let’s denote the flux through the top circular base as Φ₁ and the flux through the bottom circular base as Φ₂.
- Given that the flux through one of the bases is Φ₀, we can say:
\[ \Phi_1 = \Phi_0 \]
- Since the magnet is symmetrically placed, the flux entering through the North Pole will equal the flux exiting through the South Pole.

**Hint**: Consider the conservation of magnetic field lines. The number of lines entering one base must equal the number exiting the other.

5. **Applying Gauss's Law for Magnetism**:
- According to Gauss's law for magnetism, the total magnetic flux through a closed surface is zero:
\[ \Phi_{\text{total}} = \Phi_1 + \Phi_2 + \Phi_{\text{curved surface}} = 0 \]
- Since the curved surface does not contribute to the net flux (it is enclosed), we have:
\[ \Phi_1 + \Phi_2 = 0 \]

**Hint**: Remember that the net flux through a closed surface surrounding a magnet is zero.

6. **Finding the Flux through the Other Base**:
- From the equation above, if Φ₁ = Φ₀, then:
\[ \Phi_0 + \Phi_2 = 0 \implies \Phi_2 = -\Phi_0 \]
- Therefore, the flux through the other circular base is:
\[ \Phi_2 = -\Phi_0 \]

**Hint**: The negative sign indicates that the direction of the magnetic field through the second base is opposite to that through the first base.

### Final Answer:
The magnetic flux through the other circular base will be:
\[ \Phi_2 = -\Phi_0 \]