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Chapter 26: Problem 62

Three parallel wires of length \(l\) each carry current \(I\) in the same direction. They're positioned at the vertices of an equilateral triangle of side \(a,\) and oriented perpendicular to the triangle. Find an expression for the magnitude of the force on each wire.

Short Answer

Expert verified
The magnitude of the force on each wire is \(F = \frac{{\mu_0 I^2 l}}{{2\pi a}}\sqrt{3}\).

Step by step solution

01

Understand the principle of magnetic force

Firstly, know that according to Ampère's force law, the magnetic force \(F\) between two parallel wires of length \(l\) carrying current \(I\) and separated by a distance \(r\) is given by \[F = \frac{{\mu_0 I^2 l}}{{2\pi r}}\], where \( \mu_0 \) is the permeability of free space.
02

Determine the forces on each wire

Calculate the force \(F_1\) on one wire due to the other two wires. By symmetry, both wires exert an equal force on the third wire, and these forces make an angle of 120° with each other (since they are in an equilateral triangle). Therefore, we can calculate this force by using Ampère's law and then use vector addition to sum the forces.
03

Apply vector addition

If we denote the forces due to the two wires as \(F_1\) and \(F_2\), their magnitudes will be as follows: \[F_1 = F_2 = \frac{{\mu_0 I^2 l}}{{2\pi a}}\].The resultant force will hence be: \[F = \sqrt{{F_1^2 + F_2^2 + 2F_1F_2 \cos(120°)}} = \sqrt{{2F_1^2 (1 - \cos(120°))}}\].This simplifies to \(F = F_1\sqrt{3}\).
04

Substitute the value of \(F_1\) in step 4

Now, replace the value of \(F_1\) in this equation with the value derived from Ampère's law:\[F = F_1\sqrt{3} = \frac{{\mu_0 I^2 l}}{{2\pi a}}\sqrt{3}\].Note that due to symmetry, this force will act on each wire, pointing towards the centroid of the equilateral triangle.

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Most popular questions from this chapter

A single-turn square wire loop \(5.0 \mathrm{cm}\) on a side carries a \(450-\mathrm{mA}\) current. (a) What's the loop's magnetic dipole moment? (b) If the loop is in a uniform 1.4 -T magnetic field with its dipole moment vector at \(40^{\circ}\) to the field, what's the magnitude of the torque it experiences?

A single-turn wire loop \(10 \mathrm{cm}\) in diameter carries a 12 - A current. It experiences a \(0.015 \mathrm{N} \cdot \mathrm{m}\) torque when the normal to the loop plane makes a \(25^{\circ}\) angle with a uniform magnetic field. Find the magnetic field strength.

A 2.2 -m-long wire carrying 3.5 A is wound into a tight coil \(5.0 \mathrm{cm}\) in diameter. Find the magnetic field at its center.

Would there be a magnetic force on a piece of iron deep inside a long solenoid? Explain.

Do particles in a cyclotron gain energy from the electric field, the magnetic field, or both? Explain.
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