Go Figure 29-82 shows, in cross section, two long parallel wires spaced by distance d=10.0 cm ;

Go Figure 29-82 shows, in cross section, two long parallel...

Key Concepts

Superposition Principle

Magnetic fields are vector quantities, meaning that when multiple sources of magnetic fields are present, the net magnetic field at any point is the vector sum of the individual fields. This principle is crucial for analyzing situations where more than one current-carrying conductor contributes to the total magnetic field at a given location.

Unit Vector Notation

Unit vector notation is used to express vector quantities by indicating both magnitude and direction in a concise form. In the context of magnetic field problems, it allows the clear representation of the direction of the field components, ensuring that the final results accurately reflect both the strength and orientation of the fields.

Magnetic Field of a Long Straight Wire

The magnetic field produced by a long straight current?carrying wire is determined using the formula B = (??I)/(2?r), where ?? is the permeability of free space, I is the current, and r is the perpendicular distance from the wire. This relation is a result of applying Ampère’s Law and is fundamental for calculating magnetic fields in situations with cylindrical symmetry.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field generated by a current. When the thumb of the right hand points in the direction of the current, the curl of the fingers indicates the circular direction of the magnetic field around the wire. This rule is essential in establishing the vector direction of the field in problems involving current-carrying wires.

Transcript

00:01 In the given problem, the two straight parallel current carrying conductors are kept such that they are perpendicular to the plane of paper in the wire 1, the first conductor.

00:26 The current is coming out of the plane of paper and the direction of current in the conductor too, that is not given in the beginning.

00:38 The gap between them is given as 10 centimeter.

00:47 Then an observation point p is at a perpendicular bisector of this line joining the two conductors, such that if we join the cross sections of these two conductors with this observation point angle between these two lines is given as 90 degree so here this is 90 degree this is perpendicular to perpendicular bisector so each of these two angles will be 45 degree each here this is 45 this is 90 so clearly this angle here will also be 45 degree this distance will be half of 10 means 5 cm so now this triangle will also be an isosceles triangle triangle and see if this is 5 this distance will also be 5 so finally we can say if we consider this wire to be situated at point u this wire to be situated at point r then p q is equal to p r and that is equal to square root of 50 centimeter this is 5 this is 5 then using pythagoras theorem square root of 50 centimeter will be this hypotenuse now in the first part of the problem if current in wire 2 is outward, out of the plane of paper, then using right hand thumb rule, the directions of magnetic fields, and b2 magnetic field due to current in wire 2 at point p will be as shown in the figure so using that right -hand thumb rule considering current in wire 1 which is coming out of the plane of paper the direction of this magnetic field at point p will be like this this is b1 and due to the current in wire 2 which is also coming out of the plane of paper that will be this this is b2 and using the symmetry of these angles angle between these two magnetic fields is also equal to 90 degree now magnitudes of these two magnetic fields will be equal because the currents in both of the wires is same and the distance pq is equal p r and that magnetic field will be given by mu note by 4 pi into 2 i by r where this distance square root of 50 centimeter is equal to r now plugging in all known values here for mu not upon 4 pi this is 10 dash par minus 7 tesla meter per ampere multiplied by two times of current and current here is given as 100 ampere in both of the wires and then the distance which is square root of 50 centimeter or square root of 50 into 10 rish power minus 2 meter in tesla this will be measured so finally these two magnetic fields comes out to be 4 into 10 x to the power minus 4 divided by root 2 tesla so net magnetic field having an angle 90 degree between them if we use parallelogram law of vector addition this will be the net magnetic field along negative y -axis so again using pythagoras theorem this will be given by as a square root of b1 square plus b2 square as both the magnetic fields are same so we can say in place of b2 we can use b1 square it will be twice of b1 square having a square root or we can say this is b1 into root 2.

05:55 So finally we can say this is 4 into 10 dashpar minus 4 divided by root 2 multiplied by root 2 tesla or in vector form we can say this is as it is along minus y x is so sorry minus x x is so we can write it like minus 4 into 10 dash 4 minus 4 minus 4 minus 4 minus 4 tesla i k, which is the answer for the first part of this problem.

06:26 Now in the second part of the problem, if we reverse the direction of current in wire 2...

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