A closed curve encircles several conductors. The line integral $\oint B\cdot dl$ around this curve is $3.83\times {10}^{-4}\text{ T m}$. If you were to integrate around the curve in the opposite direction, what would be the value of the line integral? Explain.

Verified step by step guidance

1. Understand the concept of line integral of a magnetic field: The line integral of the magnetic field \( \oint \mathbf{B} \cdot d\mathbf{l} \) around a closed curve is related to the net current enclosed by the curve, according to Ampère's Law.
2. Recognize the significance of direction: The direction of integration around the curve affects the sign of the line integral. If you reverse the direction of integration, the sign of the integral will also reverse.
3. Apply Ampère's Law: Ampère's Law states that \( \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc} \), where \( I_{enc} \) is the net current enclosed by the curve. The given integral value is \( 3.83 \times 10^{-4} \) T m.
4. Consider the effect of reversing the direction: If the original integral value is positive, reversing the direction of integration will make the integral value negative, and vice versa.
5. Conclude the result: Therefore, if you integrate around the curve in the opposite direction, the value of the line integral will be \( -3.83 \times 10^{-4} \) T m.

Key Concepts

Ampère's Law

Ampère's Law relates the magnetic field around a closed loop to the electric current passing through the loop. It states that the line integral of the magnetic field B along a closed path is equal to the permeability of free space times the total current enclosed by the path. This principle is crucial for understanding how the magnetic field behaves around conductors.

Line Integral

A line integral involves integrating a vector field along a curve. In the context of magnetic fields, it calculates the total magnetic influence along a path. The direction of integration affects the sign of the result, as reversing the path direction changes the sign of the integral, which is essential for understanding the problem's requirement to integrate in the opposite direction.

Direction of Integration

The direction of integration in a line integral is significant because it determines the sign of the result. When integrating a vector field like the magnetic field, reversing the direction of the path changes the sign of the integral. Thus, integrating in the opposite direction would yield the negative of the original value, which is a key aspect of the question.