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

Knight Calc 5th Edition

Ch 29: The Magnetic Field

Problem 53

Chapter 29, Problem 53

What is the magnetic field strength at the center of the semicircle in FIGURE P29.53?

Verified step by step guidance

Step 1: Recognize that the magnetic field at the center of the semicircle is due to the current flowing through the semicircular arc. The straight sections of the wire do not contribute to the magnetic field at the center of the semicircle because their magnetic field lines are perpendicular to the center point.

Step 2: Use the formula for the magnetic field at the center of a circular loop, which is derived from the Biot-Savart law: \( B = \frac{\mu_0 I}{2R} \). For a semicircular arc, the magnetic field is half of that of a full circular loop, so the formula becomes \( B = \frac{\mu_0 I}{4R} \).

Step 3: Identify the variables in the formula: \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} \ \text{T·m/A} \)), \( I \) is the current flowing through the wire, and \( R \) is the radius of the semicircular arc.

Step 4: Substitute the given values for \( I \) and \( R \) into the formula \( B = \frac{\mu_0 I}{4R} \). Ensure that the units are consistent (e.g., current in amperes, radius in meters).

Step 5: Simplify the expression to find the magnetic field strength \( B \) at the center of the semicircle. This will give the magnitude of the field in teslas (T).

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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 direction of the magnetic field is determined by the right-hand rule, which states that if you point your thumb in the direction of current flow, your fingers curl in the direction of the magnetic field lines.

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 and the length element dl of the conductor, and inversely proportional to the square of the distance r from the current element to the point. This law is essential for calculating the magnetic field in complex geometries, such as the semicircular wire in the question.

Symmetry in Magnetic Fields

Symmetry plays a crucial role in simplifying the calculation of magnetic fields. In the case of a semicircular wire carrying current, the symmetry allows us to deduce that the magnetic field at the center of the semicircle is uniform and can be calculated using the contributions from each infinitesimal segment of the wire. This concept helps in determining the resultant magnetic field without needing to integrate over the entire shape.

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Textbook Question

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