Figure shows, in cross-section, several conductors that...

Figure shows, in cross-section, several conductors that carry currents through the plane of the figure. The currents have the magnitudes $I_{1}=4.0 \mathrm{~A}, I_{2}=6.0 \mathrm{~A}$, and $I_{3}=2.0 \mathrm{~A}$, in the directions shown. Four paths labelled $a$ to $d$, are shown. What is the line integral $\int \mathbf{B} \cdot d \mathbf{l}$ for each path? Each integral involves going around the path in the counter-clockwise direction.

Key Concepts

Ampère's Law

Ampère's Law states that the line integral of the magnetic field around a closed path is equal to the permeability of free space multiplied by the net current enclosed by the path (?B · dl = ?? I_enclosed). This fundamental law in magnetostatics provides a direct relation between the magnetic field circulating around currents and the total current passing through the surface bounded by the loop.

Enclosed Current

The concept of enclosed current involves summing the total current that passes through the area enclosed by the chosen integration path. The sign of each current contribution is determined by its direction relative to the loop’s orientation, typically following the right-hand rule. Accurately identifying the enclosed current is essential for correctly applying Ampère's Law.

Magnetic Field Line Integral

The magnetic field line integral involves integrating the component of the magnetic field tangent to a closed loop along the path of the loop. This integral quantifies the net 'circulation' of the magnetic field around the loop and, according to Ampère's Law, directly relates to the total current enclosed by the loop.

Path Orientation and Sign Conventions

The orientation of the integration path, such as counterclockwise or clockwise, determines the sign convention for the integrated magnetic field and enclosed currents. Using proper sign conventions ensures that the contributions from currents are correctly added or subtracted based on their direction relative to the integration path, which is crucial for accurate calculations using Ampère's Law.

Transcript

00:01 Hi, in the given problem there are few current carrying conductors which are enclosed within some loops.

00:09 First of all, here there is a loop a but this loop a is not having any current carrying conductor passing through it.

00:25 Now there is a loop b and this loop b is carrying a current carrying conductor passing through 8 the current i1 and that is directed into the plane of paper then there is another current carrying conductor carrying current i 2 and that is coming out of the plane of paper and now here this is loop c and finally there is a loop d involving all this loop the values of the currents are for i1 this is 4 .0 ampere which is directed into the plane of paper for i2 this is 6 .0 ampere coming out of the plane of paper and i3 also coming out of the plane of paper but having a magnitude of only 2 .0 ampere now for actually we have to find the line integrals of magnetic fields for these closed loops so starting 4 park a as no current is threading this loop so using ampere's circuit along which says line integral of magnetic field is equal to mu not times the current threading the loop but as there is no current through this loop a so the answer is 0 for the path a now for path b there is only a single current i1 reading this path so line integral of magnetic field here will be mu not times the current i1 which is having a magnitude of 4 .0 ampere so here it will be 4 .0.

02:55 0m .0 but as for the given problem the line integral involves going around counterclockwise counterclockwise but for b as the current is into the plane of paper it is means the integral is clockwise is clockwise so due to the opposite direction this line integral for path b will be minus 4 mu naught.

03:51 Now for path c this path c is including two currents i2 and i1, i2 is 6 ampere coming out i1 getting into 4 ampere.

04:07 So the more current is 6 ampere so the net current will be 6 minus 4.

04:10 So means 2 ampere and that is coming out...

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