Question:
Would the forces between wires remain the same if we altered their shape or length, or would these modifications cause significant changes to the magnetic field distribution and resulting forces?
Detailed Answer:
When we derive the standard expression for the magnetic force per unit length between two parallel current-carrying wires, we assume that:
- The wires are infinitely long
- They are perfectly straight
Under these ideal conditions, the magnetic field created by one wire is symmetric and uniform around it. The resulting magnetic force per unit length \( \frac{F}{L} \) between two wires separated by a distance \( d \), carrying currents \( I_1 \) and \( I_2 \), is given by:
\( \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d} \)
where \( \mu_0 \) is the permeability of free space.
Effect of Finite Length:
If the wires are not infinitely long, several key changes occur:
- Near the ends of the wires, the magnetic field deviates from that of an infinite wire.
- These edge effects cause the magnetic field to become non-uniform along the wire’s length.
- The simple formula above no longer accurately represents the force. Instead, one must integrate the contributions of magnetic force along the wire.
- The net force is typically less than predicted by the infinite-wire formula.
Effect of Wire Shape (Curved or Irregular):
If the wires are not straight:
- The geometry of the wire affects the magnetic field at each point in space.
- We must use the Biot–Savart Law to calculate the field:
\( \vec{B}(\vec{r}) = \frac{\mu_0}{4\pi} \int \frac{I\,d\vec{l} \times (\vec{r} – \vec{r}’)}{|\vec{r} – \vec{r}’|^3} \)
- The resulting magnetic field is no longer symmetric or uniform.
- The force on another wire must also be computed by integrating over its entire length.
- This results in a non-uniform and potentially directional force that differs from the ideal case.
Summary of Resulting Forces:
Changes in wire length and shape lead to changes in:
- Magnitude and direction of the magnetic field
- Force per unit length between the wires
For example:
- Two finite wires parallel for only a part of their length will exert force only over that portion.
- If wires are curved, the force may vary along their lengths based on local geometry.
Conclusion:
Altering the wire length or shape leads to significant changes in the magnetic field distribution. Consequently, the standard force-per-unit-length formula does not hold. Instead, one must analyze the full current configuration using principles like the Biot–Savart law to determine the actual force. These factors must be considered in practical applications involving real wires with finite size and shape.