Math Problem Statement
Solution
To solve this problem, we need to calculate the magnetic field at point due to a current flowing through an arc of radius and central angle .
Given Data:
- Current,
- Central angle,
- Radius of the arc,
Formula for Magnetic Field Due to a Circular Arc:
For a current-carrying arc, the magnetic field at the center is given by: where:
- is the permeability of free space,
- is the current,
- is the central angle in radians,
- is the radius of the arc.
Steps to Solve:
-
Convert to radians.
-
Substitute the values into the formula:
-
Simplify the equation by canceling out terms and calculating the remaining expression.
Let's calculate this value.
Calculation:
Answer:
The magnitude of the magnetic field at is approximately:
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:
- How does the magnetic field change if the radius of the arc increases while keeping other parameters constant?
- What effect would increasing the current have on the magnetic field at point ?
- How would the magnetic field at change if the central angle were doubled?
- Can we use this formula to find the magnetic field at any point not at the center? Why or why not?
- 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 .
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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)
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