The mathematical equation for magnetic field lines of force is

Answer

To find the mathematical equation for magnetic field lines of force, we can refer to Maxwell's equations, which describe the behavior of electric and magnetic fields. The specific equation that represents the magnetic field lines is derived from one of Maxwell's equations.

### Step-by-Step Solution:

1. **Understanding Magnetic Field Lines**:
Magnetic field lines represent the direction and strength of the magnetic field. They are visual representations that help us understand how magnetic fields behave in space.

2. **Maxwell's Equations**:
Maxwell's equations consist of four fundamental equations that describe how electric and magnetic fields interact. One of these equations specifically relates to magnetic fields.

3. **Divergence of Magnetic Field**:
The relevant equation for magnetic fields is given by:
\[ \nabla \cdot \mathbf{B} = 0 \]
This equation states that the divergence of the magnetic field \(\mathbf{B}\) is zero. This implies that there are no magnetic monopoles; magnetic field lines are continuous loops.

4. **Interpretation**:
The equation \(\nabla \cdot \mathbf{B} = 0\) indicates that the magnetic field lines do not begin or end at any point in space, which means they form closed loops. This is a fundamental characteristic of magnetic fields.

5. **Conclusion**:
Therefore, the mathematical equation that represents the magnetic field lines of force is:
\[ \nabla \cdot \mathbf{B} = 0 \]