Question

FIGURE EX29.37 is a cross section through three long wires with linear mass density 50 g/m. They each carry equal currents in the directions shown. The lower two wires are 4.0 cm apart and are attached to a table. What current I will allow the upper wire to 'float' so as to form an equilateral triangle with the lower wires?

Accepted Answer

Understand the problem: The goal is to determine the current $$ I  in $$the upper wire such that it 'floats' in equilibrium, forming an equilateral triangle with the two lower wires. This means the net force on the upper wire due to the magnetic forces from the lower wires and its weight must be zero. Step 1: Recall the formula for the magnetic force per unit length between two parallel current-carrying wires. The force per unit length is given by $$ F/L = \frac{\mu_0 I_1 I_2}{2 \pi d} $$, where $$ \mu_0  is $$the permeability of free space, $$ I_1 $$ and $$ I_2 $$ are the currents in the wires, and $$ d  is $$the distance between the wires. Step 2: Analyze the forces acting on the upper wire. The magnetic forces from the two lower wires will act at an angle of 60° to each other (since the wires form an equilateral triangle). The weight of the upper wire, $$ F_g = \lambda g $$, where $$ \lambda  is $$the linear mass density and $$ g  is $$the acceleration due to gravity, acts vertically downward. Step 3: Resolve the magnetic forces into components. The horizontal components of the magnetic forces from the two lower wires will cancel out due to symmetry. The vertical components will add up to balance the weight of the upper wire. Use trigonometry to express the vertical component of the magnetic force: $$ F_{vertical} = 2 \cdot F_{magnetic} \cdot \sin(60°) . $$Step 4: Set up the equilibrium condition. For the upper wire to float, the total upward magnetic force must equal the downward gravitational force: $$ 2 \cdot \frac{\mu_0 I^2}{2 \pi d} \cdot \sin(60°) = \lambda g . $$Solve this equation for $$ I $$, the current in the wires.

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Concetti chiave

Magnetic Force Between Current-Carrying Wires

Equilibrium of Forces

Linear Mass Density and Weight