Problem 13 - Fundamentals of Physics

Question

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

Step-by-Step Solution

Answer

The ratio \( \frac{L}{R} \) must be greater than 5.

Step 1: Understanding the Problem and Given Formula

The problem deals with calculating the magnetic field around a finite length wire and comparing it to an infinite wire to determine the error. The formula given for an infinite wire is: \( B = \frac{\mu_0 I}{2\pi R} \). We need to determine the ratio \( \frac{L}{R} \) for which the error is less than 1%.

Step 2: Formula for Finite Length Wire Magnetic Field

For a finite length wire of length \( L \), the formula for the magnetic field (assuming it is centered at point \( P \)) is more complex and can be represented as:\[ B = \frac{\mu_0 I}{4\pi R} \left( \sin(\theta_1) + \sin(\theta_2) \right) \]Where \( \theta_1 \) and \( \theta_2 \) are the angles from the ends of the wire to the point \( P \). When the wire is centered at \( P \), these angles can be expressed as \( \theta_1 = \theta_2 = \tan^{-1} \left( \frac{L}{2R} \right) \).

Step 3: Comparing Both Magnetic Fields

For an infinite wire, the field is \( B_{\text{infinite}} = \frac{\mu_0 I}{2\pi R} \). For the finite wire, it is:\[ B_{\text{finite}} = \frac{\mu_0 I}{2\pi R}\frac{\sqrt{1 + (\frac{L}{2R})^2}}{{\frac{L}{2R}}} \]The percentage error is:\[ \%\text{ Error} = \left| \frac{B_{\text{finite}} - B_{\text{infinite}}}{B_{\text{infinite}}} \right| \times 100\%\approx \left| 1 - \frac{\sqrt{1 + (\frac{L}{2R})^2}}{\frac{L}{2R}} \right| \times 100\% \]

Step 4: Setting up Inequality for Percentage Error

Set the percentage error to be less than 1%:\[ \left| 1 - \frac{\sqrt{1 + (\frac{L}{2R})^2}}{\frac{L}{2R}} \right| < 0.01 \] Simplify this inequality to solve for the ratio \( \frac{L}{R} \).

Step 5: Solving the Inequality

By solving the inequality, you will find:\[ \frac{\sqrt{1 + (\frac{L}{2R})^2}}{\frac{L}{2R}} \approx 1 + 0.01 \approx 1.01 \] Solving this equation will give approximately \( \frac{L}{R} > 5 \).

Step 6: Conclusion

Therefore, to ensure the percentage error is less than 1%, the ratio \( \frac{L}{R} \) must exceed 5.

Key Concepts

Infinite Wire Approximation

The concept of an infinite wire approximation is a crucial tool in understanding magnetic fields. This simplification assumes that a wire carrying current is infinitely long, which makes calculations much more manageable. For an infinite wire, the magnetic field at a distance \( R \) from the wire is given by the simple formula \( B = \frac{\mu_0 I}{2\pi R} \), where \( \mu_0 \) is the permeability of free space and \( I \) is the current.

This approximation works well when the point of interest is significantly closer to the wire than the length of the wire itself. In practical terms, this often means that if you stand very close to a long wire, the wire will appear "infinitely long" for the purposes of magnetic field calculation.

Finite Wire Effect

When dealing with real-world wires, the finite wire effect cannot be ignored. This effect comes into play because wires have a definite length, and the ends of the wire have an impact on the magnetic field calculation. For a finite wire, the magnetic field calculation becomes more complex.

The formula for a finite wire field at a midpoint \( P \), perpendicular from the wire, is:\[ B = \frac{\mu_0 I}{4\pi R} \left( \sin(\theta_1) + \sin(\theta_2) \right) \]Where \( \theta_1 \) and \( \theta_2 \) denote the angles from the ends of the wire to point \( P \). These angles depict how the field intensity spreads along the length of the wire. Thus, while calculating, it includes the deviation due to these angles.

Percentage Error in Measurements

In any scientific measurement, especially involving approximations, it's essential to understand the concept of percentage error. This measurement tells us how accurate or precise a calculation or approximation is compared to the actual values.

For magnetic field calculations with both infinite and finite wires, percentage error is calculated by determining the difference between the finite wire magnetic field \( B_{\text{finite}} \) and the infinite wire approximation \( B_{\text{infinite}} \), dividing by the infinite wire value, and then converting it to a percentage. This helps in understanding the reliability of the infinite wire approximation for specific values of \( \frac{L}{R} \), where \( L \) is the wire length and \( R \) is the distance from the wire.

Ampere's Law

Ampere's Law is a fundamental principle often used in the context of magnetic fields. This law relates magnetic fields to the electric current that produces them. It states that the line integral of the magnetic field \( B \) around a closed loop is equal to the permeability times the current enclosed in the loop: \[ \oint B \cdot dl = \mu_0 I \]

This powerful law allows for simplified calculations in symmetric situations, like our infinite wire approximation. However, because real wires are finite, the law needs modifications and additional considerations to account for endpoints, like seen in finite wire calculations.

Magnetic Field Formula

The magnetic field formula is the basis for calculating the field around conductors. For an infinite wire, it's expressed as \( B = \frac{\mu_0 I}{2\pi R} \), showing the inverse relationship between the field and the perpendicular distance \( R \) from the wire.

For finite wires, the formula becomes: \[ B = \frac{\mu_0 I}{4\pi R} \left( \sin(\theta_1) + \sin(\theta_2) \right) \]

The greater complexity reflects the fact that the finite wire's magnetic field varies with length and orientation. Understanding these formulas helps evaluate how close real-world measurements can be when using idealized models, like the infinite wire approximation, in explaining and predicting magnetic phenomena.