Shown in Fig. 15.45 is a conductor carrying a current I. The magnetic field intensity at the poi

Shown in Fig. 15.45 is a conductor carrying a current I. The...

Key Concepts

Biot–Savart Law

The Biot–Savart law provides the fundamental relationship for calculating the magnetic field produced by a small element of current. It expresses the magnetic field contribution from each segment of the conductor as proportional to the current, the length element, and the sine of the angle between the element and the position vector, and inversely proportional to the square of the distance. This law is particularly important when analyzing the magnetic field generated by conductors with nontrivial shapes.

Magnetic Field of a Circular Arc

For a circular arc carrying a current, the magnetic field at the center of curvature is derived from the Biot–Savart law and depends on the arc’s central angle. The field is directly proportional to both the current and the radian measure of the angle subtended by the arc, and inversely proportional to the radius of the arc. This concept is essential for understanding how partial sections of a full circle contribute to the net magnetic field at the center.

Superposition Principle

In electromagnetic problems, the superposition principle states that the net magnetic field created by multiple current-carrying segments is the vector sum of the magnetic fields produced by each segment individually. This principle allows complex geometrical configurations to be broken down into simpler parts whose individual contributions can be calculated and summed to determine the overall magnetic field at a given point.

Transcript

00:01 Okay, as per the question, a loop of the wire is given like this, okay? so it is, the wire is coming like this, this is the first portion, okay? then there is an arc, then it is coming like this, understand? then this goes like this, then it is coming like again in this same line, and then the arc this, then this, okay? here and it is the point o understand so this is the loop and we have to find out the magnetic field intensity at the point o okay so first of all i want to clear one thing okay that is when we meet like this this will be at this straight line okay and this is the straight line so it is pretty clear that okay due to this first it is in the line of this o okay and this second this is also in the line of the center point oh okay and this third point this is also in the line of oh and this fourth portion it is also in the line of oh understand so they will not contribute anything in the magnetic field first of all you have to clear this thing okay so the portion number one two three and four they will not contribute anything in the magnetic field at o.

01:33 Now we have to find out from the other thing.

01:38 That is, we can say the portion number, let's say it is 5, 6 and 7.

01:49 These three arcs.

01:51 So the direction, it is from here, it is here and it is here.

01:56 Okay so you can see the direction of these three arcs two are in same direction and the middle one is other direction okay so i'm pointing out the direction okay let's say this is the plus direction and this will be also plus direction so this will be the minus direction okay you can also take the direction from the right hand screw rule okay understand so this will be inside the paper and this will be inside the paper and the middle one with the out of the paper so i am taking the plus minus and plus so number five six and seven and the formula for the magnetic field at the center of an arc is equal to b equals to okay that is b equals to mu not i divided by four pi r multiplied by theta okay and the distance is given same this is r this is r and this is r understand so okay so so we can find out for the wire number 5, wire number 6 and wire number 7...

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