Two long wires lie in an x y plane, and each carries a...

Two long wires lie in an $x y$ plane, and each carries a current in the positive direction of the $x$ axis. Wire 1 is at $y=10.0 \mathrm{~cm}$ and carries $6.00 \mathrm{~A} ;$ wire 2 is at $y=5.00 \mathrm{~cm}$ and carries $10.0 \mathrm{~A}$. (a) In unit-vector notation, what is the net magnetic field $\vec{B}$ at the origin? (b) At what value of $y$ does $\vec{B}=0 ?$ (c) If the current in wire 1 is reversed, at what value of $y$ does $\vec{B}=0 ?$

Two long wires lie in an $x y$ plane, and each carries a current in the positive direction of the $x$ axis. Wire 1 is at $y=10.0 \mathrm{~cm}$ and carries $6.00 \mathrm{~A} ;$ wire 2 is at $y=5.00 \mathrm{~cm}$ and carries $10.0 \mathrm{~A}$. (a) In unit-vector notation, what is the net magnetic field $\vec{B}$ at the origin?
(b) At what value of $y$ does $\vec{B}=0 ?$ (c) If the current in wire 1 is reversed, at what value of $y$ does $\vec{B}=0 ?$

Key Concepts

Magnetic Field due to a Long Straight Conductor

This concept involves the magnetic field created by a current-carrying straight wire. According to the Biot–Savart law or Ampère’s law, the field at a distance r from an infinitely long wire carrying current I is given by B = (??I)/(2?r). This formula is fundamental when analyzing problems with current-carrying wires, especially when calculating the magnitude of the magnetic field at points in space around the wire.

Right-Hand Rule

The right-hand rule is a method used to determine the direction of the magnetic field produced by a current-carrying conductor. By pointing the thumb of the right hand in the direction of the conventional current, the curled fingers indicate the circular direction of the magnetic field lines around the wire. This tool is essential in establishing the proper vector directions for the magnetic field in various configurations.

Superposition Principle

The superposition principle states that the net magnetic field at a point due to several sources is the vector sum of the magnetic fields due to each individual source. This principle allows one to calculate complex magnetic field configurations, such as those arising from multiple wires, by considering the contributions separately and then adding them together vectorially.

Transcript

00:01 In the given problem, first of all, there are x -axis and y -axis.

00:16 Then there are two parallel current carrying conductors which are lying in x -y -plane and parallel to the x -axis.

00:29 The first one is at a distance of 10 cm from the the origin this is wire number one carrying a current i1 is equal to 6 .00 ampere then another one that is also parallel to y x xx sorry and this is named as two it is carrying a current i2 is equal to 10 .0.

01:09 Ampere first of all the gap of the second conductor from x -axis this is 5 .00 centimeter and that of the first conductor this is given as 10 .0 centimeter now in the first part of the problem if we have to find net magnetic field at the origin so for origin the distance of origin of o from the first conductor that will be 10 .0 centimeter and that of the second conductor will be 5 .00 centimeter and as the current are towards right for both the conductors hence using right hand thumb rule the directions of magnetic fields at the origin due to both the wires will be same and it will be into the plane of paper.

03:03 It will be along negative x -axis.

03:13 So the total magnetic field will be given by b1 plus d3.

03:18 Hence using the expression for the magnetic field this is mu not by 4 pi into 2 i1 by r1 plus mu not by 4 pi into 2 i2 by r2 taking this mu not by 4 pi into 2 as a common out leaving behind i 1 by r1 plus i2 by r2 now plugging in all known values this is 10 minus 7 tesla meter per ampere multiplied by 2 and for i 1 this is 6 ampere divided by r1 which is 10 centimeter or 10 into 10 x per minus 2 meter plus for i 2 this is 10 amper and for r 2 this is 5 centimeter or 5 into 10 dash per minus 2 meter finally here it becomes 2 into 10 dash per minus 5 if we take this 10 dash 4 minus 2 as a common out in denominator then it will become 10 dash 4 plus 2 in numerator so finally this is 2 into 10 dash the power minus 5 and in the bracket 10 will be the lcm and this is 6 plus 20 tesla or finally we can say this is 26 into 2.

04:50 Means this is 52 into 10 dash bar minus 6 tesla or in vector form as it is along negative x x x so this is b bar is equal to minus 5 .2 into 10 dash bar minus 10 dash bar minus 5 tesla k k.

05:25 K.

05:26 Which becomes the answer for the first part of this problem.

05:32 In the second part of the problem, here we have to find a point at which the magnetic field will be 0...

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