Calculate the magnetic field (magnitude and direction) at a...

Calculate the magnetic field (magnitude and direction) at a point $P$ due to a current $I=12.0 \mathrm{~A}$ in the wire shown in Fig. $\mathbf{P 2 8 . 6 4 .}$ Segment $B C$ is an arc of a circle with radius $30.0 \mathrm{~cm},$ and point $P$ is at the center of curvature of the arc. Segment $D A$ is an arc of a circle with radius $20.0 \mathrm{~cm},$ and point $P$ is at its center of curvature. Segments $C D$ and $A B$ are straight lines of length $10.0 \mathrm{~cm}$ each.

Calculate the magnetic field (magnitude and direction) at a point $P$ due to a current $I=12.0 \mathrm{~A}$ in the wire shown in Fig. $\mathbf{P 2 8 . 6 4 .}$ Segment $B C$ is an arc of a circle with radius
$30.0 \mathrm{~cm},$ and point $P$ is at the center of curvature of the arc. Segment $D A$ is an arc of a circle with radius
$20.0 \mathrm{~cm},$ and point $P$ is at its center of curvature. Segments $C D$ and $A B$ are straight lines of length $10.0 \mathrm{~cm}$ each.

Key Concepts

Superposition Principle

This principle states that when multiple currents or current segments contribute to the magnetic field at a point, the total magnetic field is the vector sum of the individual contributions. It enables the analysis of complex geometries by calculating the field from each segment separately and then summing them.

Magnetic Field from Finite Straight Wire Segments

This concept refers to the method for calculating the magnetic field at a point due to a finite length of a straight current-carrying conductor. The resulting expression typically involves trigonometric functions that account for the angles subtended by the endpoints of the wire segment at the point of interest.

Right-hand Rule

A mnemonic for determining the direction of the magnetic field around a current-carrying conductor. By pointing the thumb in the direction of the conventional current, the curl of the fingers indicates the orientation of the magnetic field vectors surrounding the conductor.

Biot-Savart Law

A fundamental law used to compute the magnetic field produced by an element of current. It relates the differential magnetic field generated by a small segment of current to the current's magnitude, the distance to the observation point, and the geometry of the configuration. This law is essential for calculating contributions from both straight and curved wire segments.

Magnetic Field Due to a Circular Arc

This concept involves a derived formula from the Biot-Savart Law that calculates the magnetic field at the center of curvature of an arc. The field contribution from an arc depends on the total angle subtended by the arc and the radius of the circle, which simplifies the computation in situations with circular symmetry.

Transcript

00:01 Why? in the given problem, there are a loop.

00:05 There is a loop in the form of the combination two arcs like this.

00:32 These arcs are making an angle of 120 degree at the center.

00:45 The smaller arc is having a radius smaller is equal to 20 .0 centimeter and the bigger arc is having a radius capital arc is equal to 30 .0 centimeter.

01:11 The current carried by these two arcs, the same current will be carried by them.

01:19 Whether these are the arcs or the straight segments.

01:25 So the same current will be passing through them whose magnitude is given as 12 .0 amper.

01:33 Now we have to find a net magnetic field here at the center of these two arcs, for which we will use a concept that, as we know, magnetic field at the center of a center of a circular coil is given as b is equal to mu not i by 2r and that circle if we consider this to be the circle the total angle made by the circumference at the center will be 2 pi so the magnet if you do to the complete angle means 2 pi angle is mu not i by 2 r so if we consider an arc a smaller arc making an angle theta at the center, then using unitary methods, the expression for magnetic field at the center due to arc will be given by b is equal to theta by 2 pi or we can say this is 360 degree into the normal magnetic field means mu not i by 2 r.

03:02 And then this expression will be used in order to find the magnetic field due to the two arc shaped portions, but in order to find the magnetic field due to these straight lines, we will use the concept.

03:17 We will use the biotent severed law which says the magnetic field due to the small current carrying element ideal at a point p, whose position vector is making an angle theta, that magnetic field, smaller magnetic field is given by mu -0 by 4 pi into idl sine theta by r squared.

03:44 So if that point p comes here exactly along the length of this conductor, angle theta will come out to be 0, means in that case the magnetic field will be 0.

03:53 Hence, we conclude the magnetic field at the center, see, due to these two straight segments will be zero...

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