Equation 29-4 gives the magnitude B of the magnetic field...

Equation $29-4$ gives the magnitude $B$ of the magnetic field set up by a current in an infinitely long straight wire, at a point $P$ at perpendicular distance $R$ from the wire. Suppose that point $P$ is actually at perpendicular distance $R$ from the midpoint of a wire with a finite length $L .$ Using Eq. $29-4$ to calculate $B$ then results in a certain percentage error. What value must the ratio $L / R$ exceed if the percentage error is to be less than 1.00$\% ?$ That is, what $L / R$ gives $$ \frac{(B \text { from Eq. } 29-4)-(B \text { actual) }}{(B \text { actual })}(100 \%)=1.00 \% ? $$

Equation $29-4$ gives the magnitude $B$ of the magnetic field set up by a current in an infinitely long straight wire, at a point $P$
at perpendicular distance $R$ from the wire. Suppose that point $P$
is actually at perpendicular distance $R$ from the midpoint of a
wire with a finite length $L .$ Using Eq. $29-4$ to calculate $B$ then results in a certain percentage error. What value must the ratio
$L / R$ exceed if the percentage error is to be less than 1.00$\% ?$ That
is, what $L / R$ gives
$$
\frac{(B \text { from Eq. } 29-4)-(B \text { actual) }}{(B \text { actual })}(100 \%)=1.00 \% ?
$$

Key Concepts

Error Analysis in Approximations

This concept involves comparing an approximate result obtained from an idealized model to a more accurate calculation. The percentage error quantifies the discrepancy, helping to assess the conditions under which the approximation remains valid, such as when the geometric parameters of the system fall within a certain range.

Biot–Savart Law

The Biot–Savart law is a fundamental principle used to calculate the magnetic field at a point in space due to a current element. It serves as the basis for understanding how currents create magnetic fields, and it is essential for deriving both the expressions for infinite and finite current-carrying wires.

Geometric Parameter Ratios (L/R)

The ratio of the finite wire's length to the perpendicular distance from the wire (L/R) is crucial in determining the validity of the infinite wire approximation. As this ratio increases, the finite wire behaves more like an infinite one, and the error introduced by the approximation decreases accordingly.

Infinite Wire Approximation

This concept involves using the formula for the magnetic field produced by an infinitely long current-carrying wire. In problems where symmetry simplifies the analysis, this idealization allows one to derive a simple expression for the magnetic field, which is often used as a starting point for more complex situations.

Finite Wire Magnetic Field

This concept deals with the calculation of the magnetic field generated by a wire of finite length. Unlike the infinite wire case, the integration of the Biot–Savart law over the finite extent of the current path results in additional geometric factors that account for the diminished magnetic field strength due to end effects.

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