What is the energy density at the center of a circular loop of wire carrying a 19.0-A current if the radius of the loop is 28.0 cm?

검증된 단계별 안내

Understand the problem: The energy density at the center of a circular loop of wire is related to the magnetic field generated by the current in the loop. The formula for energy density is \( u = \frac{B^2}{2\mu_0} \), where \( B \) is the magnetic field and \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} \ \text{T·m/A} \)).

Calculate the magnetic field \( B \) at the center of the loop using the formula \( B = \frac{\mu_0 I}{2R} \), where \( I \) is the current (19.0 A) and \( R \) is the radius of the loop (28.0 cm or 0.28 m).

Substitute the known values into the formula for \( B \): \( B = \frac{(4\pi \times 10^{-7})(19.0)}{2(0.28)} \). Simplify this expression to find the magnetic field at the center of the loop.

Once \( B \) is determined, substitute it into the energy density formula \( u = \frac{B^2}{2\mu_0} \). Use the value of \( \mu_0 = 4\pi \times 10^{-7} \ \text{T·m/A} \).

Simplify the expression for \( u \) to find the energy density at the center of the loop. Ensure all units are consistent throughout the calculation.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Magnetic Field Due to a Current Loop

A circular loop of wire carrying an electric current generates a magnetic field around it. The magnetic field strength at the center of the loop can be calculated using the formula B = (μ₀ * I) / (2 * R), where μ₀ is the permeability of free space, I is the current, and R is the radius of the loop. This magnetic field is crucial for understanding the energy density associated with the loop.

Energy Density in a Magnetic Field

The energy density (u) in a magnetic field is defined as the energy stored per unit volume and is given by the formula u = (B²) / (2μ₀). This concept is essential for calculating the energy density at the center of the loop, as it relates the magnetic field strength to the energy stored in that field.

Permeability of Free Space

The permeability of free space (μ₀) is a physical constant that describes how a magnetic field interacts with the vacuum of space. Its value is approximately 4π × 10⁻⁷ T·m/A. This constant is vital in the calculations involving magnetic fields and energy density, as it influences the strength of the magnetic field generated by the current in the wire.