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 $3.00 \%$ ? That is, what $L / R$ gives $$ \frac{(B \text { from Eq. } 29-4)-(B \text { actual })}{(B \text { actual })}(100 \%)=3.00 \% ? $$

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 $3.00 \%$ ? That is, what $L / R$ gives
$$
\frac{(B \text { from Eq. } 29-4)-(B \text { actual })}{(B \text { actual })}(100 \%)=3.00 \% ?
$$

Key Concepts

Biot-Savart Law

The Biot-Savart Law is a fundamental principle used to calculate the magnetic field generated by a small segment of current-carrying conductor. It involves an integration over the current distribution and directly accounts for the geometry of the current path, making it essential for analyzing the magnetic fields of conductors with finite dimensions.

Ampere's Law and Its Applicability

Ampere's Law relates the magnetic field in space to the electric current that produces it and is particularly useful in cases with high symmetry, such as infinitely long wires. However, its direct application is limited when the symmetry is broken, as in the case of finite-length wires, where a more detailed analysis is required.

Magnetic Field of a Current-Carrying Wire

The magnetic fields produced by current-carrying wires can be computed using different methods depending on the conductor's geometry. For infinitely long wires, a simple expression derived from Ampere's Law is used, while for finite wires, the integration of the Biot-Savart Law is necessary to account for the edge effects and finite length.

Error Analysis and Percentage Error

Error analysis involves quantifying the difference between an approximate solution and the exact solution. The percentage error measure is used to determine the accuracy of an approximation, such as using the infinite wire formula in a situation where the wire is actually finite. This ensures that the approximation meets a defined accuracy threshold.

Limiting Approximations in Electromagnetism

Limiting approximations involve applying formulas derived under certain assumptions, such as infinite length, to situations where those assumptions are nearly, but not exactly, met. Evaluating conditions like the ratio of the wire's length to the distance from the point of interest helps determine when these approximations are valid and how they deviate from the exact solution.

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