- - Calculate the magnetic field (magnitude and direction)...

- - 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. P28.64. 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

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field generated by a current-carrying conductor. By pointing the thumb of the right hand in the direction of the current and curling the fingers, the curled fingers indicate the direction of the magnetic field lines around the conductor. This rule is essential when determining the vector nature of the magnetic field contributions from different segments.

Magnetic Field from Straight Current Segments

The magnetic field at a point due to a finite straight current-carrying segment can be calculated using an expression that involves the sines of the angles that the endpoints of the segment subtend at the point. This result is obtained from integrating the Biot-Savart law along the straight segment, and it is critical when combining contributions from different geometries in a composite current path.

Superposition Principle

The superposition principle states that the total magnetic field produced by a current distribution is the vector sum of the magnetic fields produced by each individual current element or segment. This allows for complex current-carrying geometries to be broken down into simpler parts, whose individual magnetic field contributions can be calculated independently and then summed to yield the net magnetic field.

Biot?Savart Law

The Biot-Savart law is a fundamental principle used to calculate the magnetic field produced at a given point in space by a current element. It relates the infinitesimal contribution of the magnetic field to both the magnitude of the current element and its geometric relationship (including the distance and angle) to the point where the field is being calculated. This law serves as the basis for analyzing magnetic fields generated by various current-carrying configurations.

Circular Arc Magnetic Field Formula

For a circular arc carrying a steady current, there exists a simplified formula to calculate the magnetic field at the center of curvature. The magnetic field is directly proportional to the current and the angle subtended by the arc (in radians) and inversely proportional to the radius of the arc. This relation is derived as an integration of the Biot-Savart law over the arc and is particularly useful when the point of interest lies at the center of the arc’s circle.

Transcript

00:01 Hi, here in this given problem there are two arc shaped current carrying conductors and having two straight conductors joining these two arc shaped conductors like this.

00:36 This angle over here at point p at which we have to find the net magnetic field.

00:42 This angle is 120 degree.

00:45 This segment is a b it is named as a b c d the direction of current that is counterclockwise in the smaller segment and clockwise it will be in the larger segment and in the straight segment from dg to c and from b to a but magnetic field at point p due to the straight segments will be zero because angle made by the position vector with the current element that will be either 0 degree or 180 degree and sine 0 degree or sine 180 degree both are 0 so as per byte and sever's law magnetic field due to ab and cd will be 0 now magnetic field at point p due to the arc a d that will be coming out as current in ad is counterclockwise so this phase will start behaving like north pole and magnetic field comes out of the north pole so this is coming out and that due to the arc bc that will be getting in into the page as this direction c b this is clockwise so this face will start behaving like a south pole and magnetic field gets into the south pole and using an expression for the magnetic field at the center of a current carrying arc this is theta the sector angle by 360 multiplied by mu knot i by 2r r is the radius so using this the net magnet field will be theta which is 120 degree divided by 360 degree mu not is common by 2 that is common current i that is also common leaving behind the larger magnetic field will be due to the arc of smaller radius means a p minus the smaller magnetic field and the radius will be b p so plugging in the known values this is 120 by 360 means 1 by 3 multiplied by 4 mu not that is 4 pi multiplied by 10 to the power minus 7 multiplied by current 12 and this is by 2 and in the bracket for the radius ap that is 20 centimeter or 20 into 10 power minus 2 meter minus 30 centimeter or 30 multiplied by 10 to the power minus 2 meter...

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