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Ch 29: The Magnetic Field
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 29: The Magnetic FieldProblem 45
Chapter 29, Problem 45

Find an expression for the magnetic field strength at the center (point P) of the circular arc in FIGURE P29.45.

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Identify the relevant formula for the magnetic field produced by a current-carrying wire. For a segment of wire forming a circular arc, the magnetic field at the center is given by the Biot-Savart law: B = (μ₀Iθ)/(4πR), where μ₀ is the permeability of free space, I is the current, θ is the angle subtended by the arc in radians, and R is the radius of the arc.
Determine the angle subtended by the arc at the center, θ. If the problem specifies the angle in degrees, convert it to radians using the formula: θ_{\(\text{radians}\)} = θ_{\(\text{degrees}\)} × (π/180).
Substitute the given values for the current I, the radius R, and the angle θ into the formula B = (μ₀Iθ)/(4πR). Ensure all units are consistent (e.g., current in amperes, radius in meters, and angle in radians).
Simplify the expression to isolate the magnetic field strength B. This involves performing algebraic operations to combine constants and variables.
Verify the derived expression for correctness by checking the units. The magnetic field strength B should have units of teslas (T), which are equivalent to N/(A·m).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Magnetic Field

The magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. It is represented by the symbol B and is measured in teslas (T). The strength and direction of the magnetic field can be determined using the right-hand rule, which relates the direction of current flow to the orientation of the magnetic field lines.
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Biot-Savart Law

The Biot-Savart Law provides a mathematical description of the magnetic field generated by a current-carrying conductor. It states that the magnetic field dB at a point in space is directly proportional to the current I, the length element dl of the conductor, and the sine of the angle between dl and the line connecting the element to the point, divided by the square of the distance r from the element to the point. This law is essential for calculating the magnetic field due to complex current configurations.
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Circular Arc

A circular arc is a segment of a circle defined by two endpoints and the continuous curve between them. In the context of magnetic fields, a circular arc carrying current generates a magnetic field that can be analyzed using symmetry and the Biot-Savart Law. The magnetic field at the center of the arc is particularly significant, as it can be derived from the contributions of each infinitesimal segment of the arc, leading to a simplified expression for the total magnetic field strength.
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Related Practice
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