# Force between identical wires in a circuit (1st post)

• dtchannell88
In summary, the wires exert an attractive force on each other, and the length L can be calculated using the formula F = IL(μ0ε)/(4πa(R+rL/2)).
dtchannell88
1. Two Identical wires denoted A & B are part of an electric circuit and therefore carry some currents. The wires are characterized by resistance per unit length,r, and both have length, L, each are spaced by a distance, a. What is the magnitude, F, of the force the wires exert on each other? Is this force an attractive force? For the given resistance, R of the resistor, what should be the length, L, to achieve maximum possible, F,?

Equations i thought might be useful..:
E=I*((r*L/2) + R)

My attempt:

Since..

I =emf/2rL+R

emf=I*((r*L/2) + R)

Then the force is equal to length, by current, by magnetic field. Because the current is going in the same direction in the two wires it will be an attraction force.

When you get Force as a function of L,
F = (µ0 *E^2)/(2*pi*a*r^2*L) where
µ0=permeability
pi = 3.141592...

Im unsure if this is even right, and where I go from here.. all help is appreciated..

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sry i may have posted this in wrong section.. please close and I'll repost

A pretty comprehensive problem!
First find the current I in the relevant wires. If we find the current through the external resistor R,

$$I_{\textrm{thru R}}=\frac{\varepsilon }{R+\frac{rL}{2}}=2I$$

Now calculate the magnetic field a distance a from a wire carrying current I using Ampere's law. Use a circular amperian loop with one of the wires in question going through the center.
$$\oint_{C}\overrightarrow{B}\cdot \overrightarrow{d\l }=\mu _{0}I_{\textrm{thru C}}\Rightarrow \overrightarrow{B(a)}=\frac{\mu _{0}I}{2\pi a}\widehat{\theta }=\frac{\mu _{0}\frac{\varepsilon }{\left 2(R+\frac{rL}{2} \right )}}{2\pi a}\widehat{\theta }=\frac{\mu _{0}\varepsilon }{4\pi a(R+\frac{rL}{2})}\widehat{\theta }$$
Where the theta direction is counterclockwise looking down the current I (opposite to the I direction.)

Now let's say that's the formula for the field around the bottom wire (in your diagram, with current flowing to the right). Call the direction along the wire in the direction of current the x direction, and call the direction down the page the z direction. At the top wire, the field due to the bottom wire would be in the y direction. ($$\widehat{y}=\widehat{\theta }$$)

Then use the formula for the force on a current I due to the B field we found.
$$\overrightarrow{F}=\int I\overrightarrow{dl}\times \overrightarrow{B}=IB(a)\int_{0}^{L}dl(\widehat{x}\times \widehat{y})=IL\frac{\mu _{0}\varepsilon }{4\pi a(R+\frac{rL}{2})}\widehat{z}$$

So that's the formula for force on the top wire--the force on the bottom would be equal and opposite, so they are indeed attracting each other.

Now you can find the L satisfying a maximum F for a given R however you'd like, perhaps by setting dF/dL = 0 and solving.

Last edited:

## What is the force between two identical wires in a circuit?

The force between two identical wires in a circuit is known as the magnetic force and is caused by the interaction of the magnetic fields produced by the two wires.

## How is the force between two wires calculated?

The force between two wires can be calculated using the formula F = (μ0 * I1 * I2 * L) / (2πd), where μ0 is the permeability of free space, I1 and I2 are the currents in the two wires, L is the length of the wires, and d is the distance between them.

## What factors affect the force between two wires in a circuit?

The force between two wires is affected by the strength of the currents, the distance between the wires, and the length of the wires. It is also influenced by the permeability of the medium surrounding the wires.

## What is the direction of the force between two wires in a circuit?

The direction of the force between two wires is perpendicular to both wires and follows the right-hand rule. If the currents in the two wires are in the same direction, the force will be attractive, and if they are in opposite directions, the force will be repulsive.

## How does the force between two wires impact the overall circuit?

The force between two wires can cause the wires to move and change positions, which can affect the overall circuit and its performance. It can also cause heating and energy loss in the wires, which can impact the efficiency of the circuit.

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