A long wire carrying a 5.0 A current perpendicular to the x y -plane intersects the x -axis at x

A long wire carrying a 5.0 A current perpendicular to the x...

A long wire carrying a $5.0 \mathrm{A}$ current perpendicular to the $x y$ -plane intersects the $x$ -axis at $x=-2.0 \mathrm{cm} .$ A second, parallel wire carrying a $3.0 \mathrm{A}$ current intersects the $x$ -axis at $x=+2.0 \mathrm{cm} .$ At what point or points on the $x$ -axis is the magnetic field zero if (a) the two currents are in the same direction and (b) the two currents are in opposite directions?

Key Concepts

Magnetic Field Cancellation

Magnetic field cancellation occurs when the magnetic fields produced by different sources are equal in magnitude but opposite in direction at a specific point, resulting in a net zero field. Analyzing the conditions for cancellation involves comparing the individual field magnitudes and directions, which often requires setting up equations based on the Biot-Savart law or Ampere's law along with the superposition principle.

Superposition Principle

The superposition principle in electromagnetism states that the net magnetic field created by multiple sources is the vector sum of the individual fields produced by each source. This concept is critical when calculating the overall magnetic field at a point in space, particularly when fields from different wires interact and potentially cancel each other.

Right-Hand Rule

The right-hand rule is 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 direction in which the magnetic field circulates. This technique is essential to predict the orientation of the magnetic fields, especially when multiple wires are involved.

Magnetic Field of a Long Straight Current-Carrying Wire

This concept refers to the magnetic field produced around a wire carrying an electric current. The field strength is given by B = ??I/(2?r), where I is the current and r is the distance from the wire. The magnetic field produced by a long, straight wire forms concentric circles around the wire, and its magnitude decreases with increasing distance, a key principle when analyzing fields from multiple wires.

Transcript

00:02 Hi, in the given problem there are two parallel current carrying conductors in the direction perpendicular to xy plane.

00:17 So first of all, we show here this is xy plane, having this is x -y -axis, this is x -axis, this is y -axis.

00:32 Out of the two wires the first wire is at a distance of two centimeter let the wire be a and we are representing its cross section as the wire is perpendicular to xy plane so here we can show just it its cross section and it is given at a position of x is equal to minus 2 centimeter means it is along negative x -axis.

01:09 Another wire, let it be b, its cross -sectional area is being represented by b.

01:16 It is also perpendicular to x -y plane and it is at a position of x is equal to plus two centimeter.

01:27 Hence, this distance is also two centimeter.

01:31 So overall we can say the gap between the two wires means a, b is 4 cm.

01:38 Now, in the first part of the problem, we have to find a position where the magnetic field due to the current carried by these two wires is zero and when the currents are in the same direction.

01:56 So when the currents in the wires are in the.

02:09 The same direction and the magnitude of current is same.

02:19 It is not same.

02:21 It is here.

02:23 This is i1.

02:25 Here.

02:25 This is i2...

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