Three long wires are parallel to a z axis, and each carries...

Three long wires are parallel to a $z$ axis, and each carries a current of $10 \mathrm{~A}$ in the positive $z$ direction. Their points of intersection with the $x y$ plane form an equilateral triangle with sides of $50 \mathrm{~cm}$, as shown in Fig. $29-78 .$ A fourth wire (wire $b$ ) passes through the midpoint of the base of the triangle and is parallel to the Figure 29-78 Problem $69 .$ other three wires. If the net magnetic force on wire $a$ is zero, what are the (a) size and (b) direction ( $+z$ or $-z$ ) of the current in wire $b$ ?

Three long wires are parallel to
a $z$ axis, and each carries a current of $10 \mathrm{~A}$ in the positive $z$ direction. Their points of intersection with the $x y$ plane form an equilateral triangle with sides of $50 \mathrm{~cm}$, as shown in Fig. $29-78 .$ A fourth wire (wire $b$ ) passes through the midpoint of the base of the triangle and is parallel to the Figure 29-78 Problem $69 .$ other three wires. If the net magnetic force on wire $a$ is zero, what are the
(a) size and (b) direction ( $+z$ or $-z$ ) of the current in wire $b$ ?

Key Concepts

Magnetic Field from a Long Straight Current

A long straight current-carrying conductor produces a magnetic field that circulates around the wire; its magnitude is given by the formula B = ??I/(2?r), where I is the current and r is the perpendicular distance from the wire. This concept is essential because it describes how individual wires contribute to the magnetic field at points in space, which in turn affects forces on other currents nearby.

Force on a Current-Carrying Conductor in a Magnetic Field

A conductor carrying current in a magnetic field experiences a force that can be calculated using the formula F = I(L × B), where L is the length vector of the conductor and B is the magnetic field vector. This principle is crucial to understanding how wires interact with the magnetic fields created by other wires, determining the net force on a wire in a multi-wire system.

Superposition Principle in Electromagnetism

The superposition principle states that the net magnetic field at a point due to several current-carrying conductors is the vector sum of the magnetic fields produced by each conductor acting individually. This concept is vital when analyzing systems with multiple current sources, as it allows for the calculation of overall magnetic effects by summing the contributions from each current.

Right-Hand Rule in Magnetism

The right-hand rule is a simple mnemonic for determining the direction of the magnetic field around a current-carrying conductor or the direction of the force experienced by a conductor in a magnetic field. By aligning the thumb with the current and curling the fingers, one finds the circular direction of the magnetic field, which is essential for analyzing the interactions between currents in different directions.

Transcript

00:02 In the given problem three wires current carrying wires are kept perpendicular to the plane of paper and parallel to each other such that their cross sections taken together make an equilateral triangle as shown here so this is the first wire which is carrying current in a direction out of the paper, out of the plain of paper.

00:37 Let it be wire a.

00:38 Then there is another wire b.

00:42 This is also carrying the current outward, means out of the plan of paper.

00:47 And the third wire at the third vertex of this equilateral triangle at c.

00:54 This is also carrying current in the same direction means outward.

00:58 All the three currents ia equals to and here this is c sorry the naming this current is given as c that was c and this is d so this is c this is d and there is one more fourth current carrying conductor that is also perpendicular to the plenopaper but the direction of current is missing here the mass magnitude as well as the direction both are missing here.

01:31 Now the magnitudes of these known currents are ia is equal to ic is equal to id and that is equal to 10 ampere.

01:44 Now as currents in all three wires, a, c and d, are in the same direction.

02:11 So these wires will attract each other.

02:25 Means if you look at wire a, it will be attracted equally by the two wires c and d.

02:34 So here this is the force of attraction exerted at a by c...

Please give Ace some feedback