Two long, straight wires, each with a current of 5.0 A, are placed on two corners of an equilate

step 1 Hi, in the given problem, two parallel current carrying conductors. These are arranged perpendic

Two long, straight wires, each with a current of 5.0 A, are...

Key Concepts

Right-Hand Rule for Magnetic Field Direction

The right-hand rule is a mnemonic used to determine the direction of the magnetic field around a current-carrying conductor. By pointing the thumb of the right hand in the direction of the current, the curl of the fingers shows the circular path of the magnetic field lines around the wire. This technique helps in understanding and visualizing the orientation of the magnetic field vectors in space.

Lorentz Force on a Moving Charge

A moving charged particle in the presence of a magnetic field experiences a force known as the Lorentz force, given by F = q(v × B). Here, q is the charge, v is the velocity vector, and B is the magnetic field vector. The direction of this force is determined by the cross product, meaning it is perpendicular to both the velocity of the charge and the magnetic field, which is instrumental in analyzing the dynamics of charged particles in electromagnetic fields.

Magnetic Field from a Long, Straight Conductor

The magnetic field produced by a long, straight current-carrying wire is found using the relation B = ??I/(2?r), where ?? is the permeability of free space, I is the current, and r is the radial distance from the wire. This concept is fundamental in electromagnetism because it allows us to calculate the field’s magnitude generated by a constant current along a straight path.

Superposition Principle in Magnetism

Magnetic fields obey the superposition principle, meaning that the total magnetic field at a point is the vector sum of the fields produced by each individual current source. This concept is crucial when multiple wires or magnetized objects contribute to the net field, as their individual fields must be added together taking both magnitude and direction into account.

Transcript

00:01 Hi, in the given problem, two parallel current carrying conductors.

00:06 These are arranged perpendicular to the plane of paper.

00:12 One of the conductor is carrying current which is coming out of the plane of paper and another conductor is carrying the current which is getting into the plane of paper.

00:27 And they are arranged such as to be at the two corners of an equilateral triangle.

00:39 And here this is the third corner of that equilateral triangle.

00:48 Suppose this is a point p.

00:54 Now, both the conductors are carrying same current through them having the magnitude i1 is equal to i 2 is equal to 5 .0 ampere and the side length of this equilateral triangle is given as 3 .2 centimeter now we have to find the net magnetic field at the vertex the top vertex of this equilateral triangle at point p.

01:32 So as the currents in the conductors are same and the distance of observation point these distances will also be same being the sides of an equilateral triangle hence we can say the magnitudes of the magnetic field at the vertex due to these two conductors will be same and that will be given by the expression mu not upon four pi into two to i by r, where r will be taken as the side of this equilateral triangle.

02:06 Now if we look for the direction, directions of these two magnetic fields using right -hand thumb rule, the direction of magnetic field due to the conductor, this first conductor, here it will be like this, perpendicular to the line joining, the observation point to this conductor this is b1 due to the current passing through conductor 1 and if this conductor is 2 magnetic field due to it will be perpendicular to the line joining this observation point to the conductor 2 and here it will be named as b2 now as these two angles are 90 degrees each this is 60 degree so if we add these three angles and subtract them from 360 degree this angle between two magnetic fields here will come out to be 120 degree it is necessary to find this angle in order to find net magnetic field which will be given by the vector sum of these two fields and and definitely this net magnetic field here will come out to be upward.

03:32 In the plane of paper means we can say this is along y -axis.

03:38 Now first of all, we find the magnitude of these two magnetic fields for which we plug in all known values.

03:47 This is 10 -dish -par minus 7 tesla meter per ampere multiplied by 2 times of current which is 5 .0 amp.

03:57 Divided by the distance 3 .2 centimeter or 3 .2 into 10 dash bar minus 2 meter.

04:07 Then canceling ampere with the ampere meter with the meter finally these two magnetic fields comes out to be 3 .125 into 10 dash for minus 5 tesla each.

04:31 So, as a the angle between them we have discussed it in the figure that is 120 degree so net magnetic field at the observation point will be given by using triangle of a factor addition this is square root of b1 is b2 square plus twice of b1 into b2 into cost theta so now putting all the known values of b1 and b2 and cost theta as 10 dash per minus 5 will be common with all of them...

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