Figure 29-85 shows, in cross section, two long parallel...

Figure $29-85$ shows, in cross section, two long parallel wires that are separated by distance $d=18.6 \mathrm{~cm} .$ Each carries $4.23 \mathrm{~A}$ out of the page in wire 1 and into the page in wire $2 .$ In unitvector notation, what is the net magnetic field at point $P$ at distance $R=34.2 \mathrm{~cm},$ due to the two currents?

Key Concepts

Magnetic Field due to a Long, Straight Current-Carrying Wire

This concept involves the calculation of the magnetic field produced by a current flowing through an infinitely long, straight wire. The magnetic field's magnitude is determined by the relation B = (??I)/(2?r), where I is the current and r is the perpendicular distance from the wire. The field circulates around the wire in closed loops, and this expression is a direct result derived from Ampère’s Law and the Biot–Savart Law.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field created by a current-carrying wire. By pointing the thumb of the right hand in the direction of the current, the curl of the fingers shows the circular direction of the magnetic field lines. This rule is essential for visualizing and correctly assigning vector directions in magnetic field problems.

Superposition Principle

The superposition principle states that in linear systems, the net response (in this case, the net magnetic field) at a point is the vector sum of the individual responses from each source. For multiple current-carrying wires, the magnetic field at any point is obtained by calculating the field due to each wire separately and then summing these fields vectorially.

Vector Representation of Magnetic Fields and Unit Vector Notation

In electromagnetism, fields are vector quantities that have both magnitude and direction. Representing the magnetic field in unit vector notation allows for clear and unambiguous depiction of its components along defined axes. This approach is essential when combining fields from different sources to obtain the net field, as it ensures that directional components are correctly accounted for.

Transcript

00:02 In this given problem there are two parallel current carrying wires.

00:08 Suppose this wire 1 it is kept at a point a and another wire which is parallel to the first one.

00:20 Wire 2 suppose it is kept at a point b.

00:24 Current in the wire 1 it is coming out of the plane of paper perpendicularly and current in wire 2 that is getting into the plane of paper perpendicularly.

00:36 The gap between these two wires that is given as 18 .6 centimeter.

00:46 Now at their perpendicular bisector there is the observation point p at which we have to find net magnetic field.

01:01 If you consider this midpoint to be c, then c p, which is given as r here, this distance r is having a value of 34 .2 centimeter and current in both of these wires in wire 1 and wire 2 means i1 is equal to i2 in magnitude and that is given as 4.

01:35 Point two three am here now using right hand thumb rule directions of magnetic fields at point b to i1 and i2 means the magnetic fields are b1 and b2 and these directions are as shown in the figure and here for wire one using right -hand thumb rule this magnetic field will be perpendicular to the line joining the wire with the observation point is perpendicular to ap like this so this is the magnetic field be one and now similarly magnetic field due to the wire put at point b using again the same right -hand thumb rule this magnetic field here this is perpendicular to the line b -p and as the current in these two wires are same and the distance ap will be equal to bp so these two magnetic fields will be having same magnitude but to find that magnitude of these two magnetic fields we should have the magnitude of this distance whether it is ap or b p this this ap or b p that distance will be given by pythagoras theorem as a root of ac square plus cp square here this ac will be half of 18 .6 centimeter...

Please give Ace some feedback