Two very long, straight, parallel wires carry currents that...

Two very long, straight, parallel wires carry currents that are directed perpendicular to the page, as in Figure P30.9. Wire 1 carries a current $I_{1}$ into the page (in the - z direction) and passes through the $x$ axis at $x=+a$ . Wire 2 passes through the $x$ axis at $x=-2 a$ and carries an unknown current $I_{2}$ . The total magnetic field at the origin due to the current-carrying wires has the magnitude $2 \mu_{0} I /(2 \pi a) .$ The current $I_{2}$ can have either of two possible values. (a) Find the value of $I_{2}$ with the smaller magnitude, stating it in terms of $I_{1}$ and giving its direction. (b) Find the other possible value of $I_{2}$ .

Key Concepts

Magnetic Field Due to a Long, Straight Conductor

This concept explains that a long, straight current-carrying conductor produces a magnetic field whose magnitude at a distance from the wire can be calculated using Ampère's law or the Biot–Savart law. The resulting expression shows that the field decreases inversely with distance from the wire, and its direction encircles the wire following the right-hand rule.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field around a current-carrying conductor. When the thumb of the right hand is pointed in the direction of conventional current flow, the curled fingers represent the direction in which the magnetic field lines circulate around the wire.

Superposition of Magnetic Fields

This principle states that when multiple current-carrying wires are present, the total magnetic field at any point is the vector sum of the individual magnetic fields created by each wire. Understanding this concept is essential when calculating the net magnetic field from separate contributions, as it involves both magnitude and direction.

Solving for Unknown Currents in Magnetic Field Equations

In problems where the net magnetic field is known and contributions from multiple current-carrying wires are superimposed, setting up and solving equations based on the known field magnitudes allows for the determination of unknown current values. This involves applying both the magnitude formula for each wire's magnetic field and considering their directional contributions based on their positions.

Transcript

00:01 In the given problem, here this is the x -axis.

00:06 This is the origin, then at a distance of a along the x -axis, there is a current carrying conductor carrying current i -1 along negative z -axis into the plane of paper.

00:27 And at a distance of 2a there is another current carrying conductor carrying a current i2 but the direction of current and the magnitude of this current i2 is not known.

00:44 The magnetic field at the origin has been given as b o is equal to mu not by 2 pi into 2 i 1 by a.

00:55 So we have to find the two possible values of current i2 in order to have this much current at the origin.

01:03 So for this problem, we will have to use the biotent severed law.

01:10 Using biotent severed law, the expression for magnetic field due to a straight current carrying conductor is given by mu not by 4 pi into 2i by r, where r is the distance of that observation point from the straight current carrying conductor.

01:28 So in the first part of the problem, we have to find the smallest possible value of current i2, which will give the net magnetic field at origin as given here.

01:41 So for the smallest possible value of i2, the two magnetic fields should be added.

02:03 So the two magnetic fields at origin because of these two wires will be added only when the two currents are in opposite direction using right hand thumb rule.

02:14 As the direction of magnetic field due to this i1 is along negative y -axis.

02:23 It is along positive y -axis using right -hand thumb rule.

02:30 So the direction of magnetic field due to this i2 should also be along.

02:35 Positive y -axis and this is only possible when the current i2 should be in positive z direction...

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