A long, straight wire lies along the y -axis and carries a...

A long, straight wire lies along the $y$ -axis and carries a current $I=8.00 \mathrm{~A}$ in the $-y$ -direction (Fig. E28.19). In addition to the magnetic field due to the current in the wire, a uniform magnetic field $\overrightarrow{\boldsymbol{B}}_{0}$ with magnitude $1.50 \times 10^{-6} \mathrm{~T}$ is in the $+x$ -direction. What is the total field (magnitude and direction) at the following points in the $x z$ -plane: (a) $x=0, z=1.00 \mathrm{~m} ;$ (b) $x=1.00 \mathrm{~m}$ $z=0 ;(\mathrm{c}) x=0, z=-0.25 \mathrm{~m} ?$

A long, straight wire lies along the $y$ -axis and carries a current $I=8.00 \mathrm{~A}$ in the $-y$ -direction (Fig. E28.19). In addition to the magnetic field due to the current in the wire, a uniform magnetic field $\overrightarrow{\boldsymbol{B}}_{0}$ with magnitude $1.50 \times 10^{-6} \mathrm{~T}$ is in the $+x$ -direction. What is the total field (magnitude and direction) at the following points in the $x z$ -plane:
(a) $x=0, z=1.00 \mathrm{~m} ;$ (b) $x=1.00 \mathrm{~m}$
$z=0 ;(\mathrm{c}) x=0, z=-0.25 \mathrm{~m} ?$

Key Concepts

Magnetic Field from a Long Straight Current-Carrying Wire

This concept pertains to the magnetic field generated by a steady current flowing through a long, straight conductor. The field is given by the formula B = (??I)/(2?r), where I is the current, r is the perpendicular distance from the wire, and ?? is the permeability of free space. This formula applies under the assumption of an infinitely long wire, providing a field that decreases inversely with distance from the wire.

Right-Hand Rule

The right-hand rule is used to determine the direction of the magnetic field surrounding a current-carrying conductor. When you curl the fingers of your right hand in the direction of current flow, your thumb points along the direction of the current, and the curl of your fingers indicates the circular orientation of the magnetic field lines around the wire.

Uniform Magnetic Field

A uniform magnetic field is a magnetic field that has the same magnitude and direction at every point in space. In problems involving multiple fields, such as one from a current-carrying wire and one external uniform field, understanding the properties of a uniform field helps in determining how it contributes to the net magnetic field at various points.

Superposition Principle

The superposition principle states that the net magnetic field produced by multiple sources is the vector sum of the individual fields. This principle allows us to calculate the total field at any point by considering the contributions from each source separately and then combining them as vectors, taking into account both magnitude and direction.

Transcript

00:01 Hi, in the given problem there is y -axis which is shown vertically downward.

00:10 Then there are the x and xx also.

00:17 Here this is the x -axis shown like this and this is the x -axis.

00:27 Supposed to be coming out perpendicularly from the plane of paper here like this.

00:33 So this is x -axis, this is x -axis plus x -axis, this is plus x -axis, this is plus x -axis, this is plus y, and here this will be minus y.

00:49 An infinite length wire is placed along this y -axis which is carrying a current i having a magnitude of 8 .00 ampere there is a magnetic field an external magnetic field along positive x -axis which is given as v0 is equal to 1 .50 into 10 dash to the power minus 6 tesla now we have to find that net magnetic field due to this wire and due to this external magnetic field the sum of these two magnetic fields at various given points so first of all in the first part of the problem the observation point is situated here at which x is equal to 0 and z has been given as 1 .00 meter means if we consider this to be the origin the distance of observation point from this current carrying conductor is 1 .0 meter so using biotent severed's law magnetic field due to wire b w will be given by the expression mu not by 4 pi into 2 i by r so plugging in all known values this is 10 dash 1 minus 7 tesla meter per ampere into 2 into 8 tamp here divided by r which is 1 .0 meter.

02:23 So finally here this magnetic field comes out to be 1 .6 into 10 tish to power minus 6 tesla.

02:31 And using right hand thumb rule, the direction of this magnetic field is along negative x axis like this here.

02:50 So this is b w in first part of this problem.

02:57 Here we can mark it as part a...

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