Figure 29-45 shows two current segments. The lower segment...

Figure $29-45$ shows two current segments. The lower segment carries a current of $i_{1}=0.40 \mathrm{~A}$ and includes a semicircular arc with radius $5.0 \mathrm{~cm},$ angle $180^{\circ},$ and center point $P .$ The upper segment carries current $i_{2}=2 i_{1}$ and includes a circular arc with radius $4.0 \mathrm{~cm},$ angle $120^{\circ},$ and the same center point $P$ What are the (a) magnitude and (b) direction of the net magnetic field $\vec{B}$ at $P$ for the indicated current directions? What are the (c) magnitude and (d) direction of $\vec{B}$ if $i_{1}$ is reversed?

Figure $29-45$ shows two current segments. The lower segment carries a current of $i_{1}=0.40 \mathrm{~A}$ and includes a semicircular arc with radius $5.0 \mathrm{~cm},$ angle $180^{\circ},$ and center point $P .$ The upper segment carries current $i_{2}=2 i_{1}$ and includes a circular arc with radius $4.0 \mathrm{~cm},$ angle $120^{\circ},$ and the same center point $P$ What are the (a) magnitude and (b) direction of the net magnetic field $\vec{B}$ at $P$ for the indicated current directions? What are the
(c) magnitude and (d) direction of $\vec{B}$ if $i_{1}$ is reversed?

Key Concepts

Right-Hand Rule

The right-hand rule is a mnemonic that helps determine the direction of the magnetic field produced by a current-carrying conductor. By pointing the thumb in the direction of current flow, the curling of the fingers indicates the circular direction of the magnetic field lines. This rule is crucial when analyzing the directions of fields, especially for curved segments, and for understanding changes in field direction when current directions are reversed.

Superposition Principle

This principle states that the total magnetic field created by multiple current-carrying conductors is the vector sum of the individual magnetic fields each conductor would produce on its own. It is particularly important when dealing with complex configurations, allowing the calculation of net effects by adding the contributions from each segment separately.

Biot-Savart Law

This fundamental law provides the basis for calculating the magnetic field produced by a current element. It relates the magnetic field contribution from a small segment of current to the magnitude, direction, and distance from that segment. The law is essential for understanding how current distributions, including segments of arcs, contribute to the overall magnetic field at a point.

Magnetic Field of a Circular Arc

A circular arc carrying a current produces a magnetic field at the center of curvature that is proportional to the current, the angle subtended by the arc (in radians), and inversely proportional to the radius of the arc. The formula B = (?? i ?)/(4? R) summarizes this relationship and is widely used to determine the magnetic field contributions from segments of loops or arcs.

Transcript

00:01 Hi, in the given problem, there are two segments in arc shape, segments of the current carrying wire.

00:11 The first one is in the form of a semicircle like this.

00:23 This is the center of the semicircle and this center is shared by one more arc here.

00:35 This arc is making an angle.

00:38 This is a smaller arc and it is making an angle of 120 degree.

00:46 At the center which is shared by both of these loops.

00:51 The center is p.

00:53 The current here passing through the lower arc, semi -circular arc, this is i -1 and passing through the upper arc, this is i -2.

01:01 Here is this angle theta which whose value is 120 degree current i1 is having a magnitude 0 .40 amper radius of the bigger loop that is 5 .0 cm current in the smaller loop upper loop that is i2 is equal to twice of i1 which will become 0 .80amper.

01:29 Here and its radius here is that is a smaller loop having a radius equal to 4 .0 centimeter now in the first part of the problem we have to find net magnetic field at the common center along with its direction so first of all b1 means magnetic field due to the semicircular arc at the center it will be given by mu not i1 divided by 4 r1.

02:01 So plugging in known values here, 4 pi into 10 dash per minus 7, tesla meter per ampere for mu not into i1, which is 0 .40 ampere divided by 4 times of r, r1, which is 5 cm or 5 into 10 dash per minus 2 meter.

02:20 So this magnetic field here will come out to be 0 .25 into 10 dash per minus 5, tesla, or or we can say this is 2 .5 micro tesla and its direction will be as the current is in clockwise direction sorry counterclockwise direction so this face will start behaving like a north pole and the magnetic field comes out of the north pole so this magnetic field will be outward similarly, the magnetic field due to the smaller loop and that will be given by theta by 360 degree times the magnetic field due to the complete loop, mu not i2 divided by 2 r2.

03:09 So plugging in the known values here for theta this is 120 degree divided by 360 degree, mu not 4 pi into 10 dash power minus 7 tesla meter per amp here again for current i2 this is 0 .80 ampere divided by 2 times of radius which is 0, which is 4 cm or 4 into 10 .m...

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