(II) Consider a straight section of wire of length d, as in...

(II) Consider a straight section of wire of length $d,$ as in Fig. $28-48,$ which carries a current $I .(a)$ Show that the magnetic field at a point $P$ a distance $R$ from the wire along its perpendicular bisector is $$ B=\frac{\mu_{0} I}{2 \pi R} \frac{d}{\left(d^{2}+4 R^{2}\right)^{\frac{1}{2}}} $$ (b) Show that this is consistent with Example $28-11$ for an infinite wire

(II) Consider a straight section of wire of length $d,$ as in Fig. $28-48,$ which carries a current $I .(a)$ Show that the magnetic field at a point $P$ a distance $R$ from the wire along its perpendicular bisector is
$$
B=\frac{\mu_{0} I}{2 \pi R} \frac{d}{\left(d^{2}+4 R^{2}\right)^{\frac{1}{2}}}
$$
(b) Show that this is consistent with Example $28-11$ for an infinite wire

Key Concepts

Limiting Behavior and Consistency Checks

Verifying that the magnetic field expression for a finite wire converges to the known result for an infinite wire, as its length increases, is a crucial consistency check in physics. This involves taking the limit of the derived expression as the wire length becomes very large and ensuring that it matches the expression derived from Ampere's Law for an infinite current-carrying wire. This approach solidifies the understanding that different methods of deriving magnetic fields are consistent with each other.

Magnetic Field of an Infinite Wire

In the special case of an infinite wire, symmetry allows the use of Ampere's Law to derive a more straightforward expression for the magnetic field, which is inversely proportional to the distance from the wire. The infinite wire serves as an idealized model that provides a benchmark for understanding how magnetic fields behave in simpler, highly symmetric conditions. The finite wire expression should reduce to this well-known result in the limit where the wire length tends to infinity.

Biot-Savart Law

The Biot-Savart Law is a fundamental principle used to calculate the magnetic field produced by a small segment of current. It states that the magnetic field contribution from a current element is proportional to the current, the length of the segment, and the sine of the angle between the element and the line joining it to the observation point, while being inversely proportional to the square of the distance. This principle is essential for deriving the magnetic field at any point due to finite current-carrying wires by integrating over the wire’s length.

Magnetic Field of a Finite Current-Carrying Wire

For a finite wire, the magnetic field is calculated by integrating the contributions from each infinitesimal segment along the current-carrying path. The geometry, such as the location of the observation point relative to the wire (like being on the perpendicular bisector), simplifies the integration by exploiting symmetry. This results in an expression that includes the length of the wire and the distance from the wire, demonstrating how finite dimensions affect the magnetic field.

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