(III) A square loop of wire, of side d, carries a current I. (a) Determine the magnetic field B at points on a line (call it the 𝓍 axis) perpendicular to the plane of the square which passes through the center of the square (Fig. 28–56). Express B as a function of 𝓍, the distance from the center of the square. (b) For 𝓍 ≫ d, does the square appear to be a magnetic dipole? If so, what is its dipole moment?

검증된 단계별 안내

Step 1: Begin by understanding the geometry of the problem. The square loop of wire has a side length 'd' and carries a current 'I'. The goal is to calculate the magnetic field 'B' at points along the 𝓍-axis, which is perpendicular to the plane of the square and passes through its center.

Step 2: Use the Biot-Savart law to calculate the magnetic field contribution from each segment of the square loop. The Biot-Savart law states:

B = μ

₀I4π∫dl×rr3. Here, 'dl' is the infinitesimal length of the wire, 'r' is the distance vector from the wire to the point of interest, and μ₀ is the permeability of free space.

Step 3: Recognize the symmetry of the square loop. Due to the symmetry, the contributions to the magnetic field from opposite sides of the square will partially cancel out in certain directions. Focus on the components of the field along the 𝓍-axis, as other components will cancel due to symmetry.

Step 4: For part (b), analyze the behavior of the magnetic field when 𝓍 ≫ d. In this limit, the square loop can be approximated as a magnetic dipole. The magnetic dipole moment 'm' is given by:

m = I \times A

, where 'A' is the area of the square loop. Since the area of the square is

d^{2}

, the dipole moment becomes

m = I \times d^{2}

Step 5: Conclude by noting that for large distances (𝓍 ≫ d), the magnetic field of the square loop resembles that of a magnetic dipole. The field can be expressed using the dipole field formula:

B = μ

₀m4πx3, where 'm' is the dipole moment and 'x' is the distance from the center of the square.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Magnetic Field Due to a Current Loop

The magnetic field generated by a current-carrying loop of wire can be calculated using the Biot-Savart law. This law states that the magnetic field at a point in space is proportional to the current flowing through the wire and inversely proportional to the square of the distance from the wire to the point. For a square loop, the magnetic field can be derived by integrating contributions from each segment of the loop.

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that represents the strength and orientation of a magnetic source. For a current loop, it is defined as the product of the current flowing through the loop and the area of the loop. In the case of a square loop, the dipole moment can be calculated as μ = I * A, where A is the area of the square, which is d² for a square of side length d.

Limit of Large Distance (𝓍 ≫ d)

When analyzing the magnetic field at points far away from a current loop (where the distance 𝓍 is much greater than the dimensions of the loop, d), the loop can be approximated as a magnetic dipole. In this limit, the magnetic field behaves similarly to that of a point dipole, allowing for simplifications in calculations and leading to the conclusion that the field decreases with distance as 1/𝓍³.