Math Problem Statement

A current path shaped as shown in the figure produces a magnetic field at P, the center of the arc. If the arc subtends an angle of θ = 35.0° and the radius of the arc is 0.700 m, what are the magnitude and direction of the field produced at P if the current is 5.00 A?

Solution

To solve this problem, we need to calculate the magnetic field BB at point PP due to a current II flowing through an arc of radius rr and central angle θ\theta.

Given Data:

  • Current, I=5.00AI = 5.00 \, \text{A}
  • Central angle, θ=35.0=35.0180π=35π180radians\theta = 35.0^\circ = \frac{35.0}{180} \pi = \frac{35 \pi}{180} \, \text{radians}
  • Radius of the arc, r=0.700mr = 0.700 \, \text{m}

Formula for Magnetic Field Due to a Circular Arc:

For a current-carrying arc, the magnetic field BB at the center is given by: B=μ0Iθ4πrB = \frac{\mu_0 I \theta}{4 \pi r} where:

  • μ0=4π×107Tm/A\mu_0 = 4\pi \times 10^{-7} \, \text{T} \cdot \text{m}/\text{A} is the permeability of free space,
  • II is the current,
  • θ\theta is the central angle in radians,
  • rr is the radius of the arc.

Steps to Solve:

  1. Convert θ\theta to radians. θ=35π1800.6109radians\theta = \frac{35 \pi}{180} \approx 0.6109 \, \text{radians}

  2. Substitute the values into the formula: B=(4π×107)×5.00×0.61094π×0.700B = \frac{(4\pi \times 10^{-7}) \times 5.00 \times 0.6109}{4 \pi \times 0.700}

  3. Simplify the equation by canceling out π\pi terms and calculating the remaining expression.

Let's calculate this value.

Calculation:

B=(4×107)×5.00×0.61094×0.700B = \frac{(4 \times 10^{-7}) \times 5.00 \times 0.6109}{4 \times 0.700} B=(2×106)×0.61090.700B = \frac{(2 \times 10^{-6}) \times 0.6109}{0.700} B1.744×106T=1.744µTB \approx 1.744 \times 10^{-6} \, \text{T} = 1.744 \, \text{µT}

Answer:

The magnitude of the magnetic field at PP is approximately: B1.744µTB \approx 1.744 \, \text{µT}

The direction of the magnetic field follows the right-hand rule, which, given the current's orientation, would point into or out of the plane of the arc depending on the specified direction in the diagram.

Would you like further clarification on any part of this solution, or details on related concepts?

Here are some follow-up questions that might deepen your understanding:

  1. How does the magnetic field change if the radius of the arc increases while keeping other parameters constant?
  2. What effect would increasing the current have on the magnetic field at point PP?
  3. How would the magnetic field at PP change if the central angle θ\theta were doubled?
  4. Can we use this formula to find the magnetic field at any point not at the center? Why or why not?
  5. What would happen to the magnetic field if the arc were part of a complete circle?

Tip: For arcs covering less than a full circle, only the central angle fraction of the full circle's field contributes, making the calculation proportional to θ\theta.

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Math Problem Analysis

Mathematical Concepts

Electromagnetism
Magnetic Field of a Current-Carrying Wire
Angles and Radians

Formulas

B = (μ₀ I θ) / (4 π r)
θ in radians = θ in degrees × π / 180
μ₀ = 4π × 10⁻⁷ T·m/A

Theorems

Biot-Savart Law

Suitable Grade Level

Grades 11-12 (Physics)