# Magnetic Field Between Two Parallel Wires (9 Facts )

We are familiar with the interaction of the magnetic field with various materials. This post will briefly note the magnetic field between two parallel wires.

Current carrying wires are associated with the magnetic field because of the movement of the charges inside the wire. So magnetic fields are always influenced by the characteristics of the current inside the wire. If two parallel wires are placed, the behavior of the magnetic field due to both wires and the facts influencing the magnetic field between two parallel wires is given in this post.

## What is the magnetic field between two wires?

When two wires carrying current are placed parallel, both wires are intended to produce a magnetic field of equal magnitude. Both the field combined to form a single uniform field. These fields are due to the motion of the charges carrying current inside the wire.

The magnetic field between two parallel wires follows the famous right-hand rule. The thumb of your right hand will be in the direction of the conventional current, and all other fingers are curled, indicating the magnetic field encircled around the wire. So if two parallel wires carry current encircled by magnetic fields around them, the magnetic field intersects at some point. The behavior magnetic field thus generated by the parallel wires follows two cases;

A detailed explanation regarding the above mentioned cases is provided in the following section.

## How to find the magnetic field of two parallel wires?

Generally, the magnetic field in a wire can find out by using the formula,

Where μ0 is the permeability of the free space, its value is 4π×10-7 Tesla, I is the current flowing across the wire, and r is the dimension of the wire.

Since we are talking about the magnetic field between two parallel wires, by modifying the same equation, we can easily calculate the magnetic field between two parallel wires if we know the current flowing across both wires and the dimension of both wires.

For example, consider two wires carrying current I1 and I2. The separation between the wires and the field of both wires is r1 and r2; the magnetic field is generated around both wires. The magnetic fields of both wires will be B1 and B2.

The magnetic field B1 of the first wire is,

The magnetic field B2 of the second wire is,

Since we are finding the magnetic field between two parallel wires, the difference between B1 and B2 gives the required magnetic field between both wires, as B=B1-B2

If the distance between the wire and the point at which magnetic field is measured of both the wires are the same, i.e., r1=r2=r, then the equation of magnetic field between two parallel wires is given as

## What happens when a wire is parallel to a magnetic field?

If a current carrying wire is kept parallel to the magnetic field, the force acting between the wire and magnetic field becomes zero. This is because force depends on the direction of current and the magnetic field; force is equal to the sin of the angle between them. So when the wire is parallel to a magnetic field, the angle between the current flow and magnetic field is either 0° or 180°. Thus, force is also zero.

Generally, magnetic field due to any current tends to create magneto-motive force orthogonal external field. The magneto-motive force thus produced flows normally to the current, increases the density field lines, and tries to get close to the wire to intersect the current. But when the wire is parallel to the field, the affinity is zero, no magneto-motive force is created to increase the density, and the field never intersects the current.

This is given by the equation Fm=|v||B|sinθ

When current is parallel to a magnetic field, the angle between current and field is 0 or 180. For both, sin(0°)=sin(180°)=0.

Fm=|v||B|(0)

Fm=0.

This shows the force is zero when the current is parallel to the magnetic field.

## Where is the magnetic field between two wires zero?

At the midway between the two wires, if the flow of current in an individual wire is in the same direction, the magnetic field will be zero. This action is because the wire-carrying current acts as a giant magnet.

At the midpoint, zero current will flow across the center of the wire; thus, the charges become stationary at the center of both wires since we know that the static charges cannot produce a magnetic field. Thus, the magnetic field is zero at the midpoint between the two parallel wires.

When the current flowing in both wires is in the same direction, magnetic fields generated in both magnets offset at the center, and both wires tend to move closer. So the magnetic fields cancel out.

## What is the force between two parallel wires?

The ordinary current generates a magnetic field in the wire to create force. Similarly, magnetic fields are generated around the wires when two current-carrying wires are parallel, which exerts some force. The force thus created between two wires defines the fundamental concept of ampere.

For example, let two wires, A and B, are separated by distance r, and both wires carry the currents I1 and I2, and both produce the magnetic field B1 and B2, respectively.

The magnetic field B1 at the wire is given by

The field produced by wire A exerts a certain force on wire B. The force F due to wire A on B is given by

F=IBL sinθ; the value of sinθ=1 because the force exerted is perpendicular to both field and current.

F=I2LB1; where l is the length of the wire B.

From the first equation; substituting the value of B1, we get,

On rearranging the equation, we get

F/L represents the force per unit length along the wire that gives the ampere.

The force is attractive if the current flowing across the wires is in the same direction. If the current flows in the opposite direction, the force is repulsive. The force between two parallel wires is independent of the current. This force exists even if there is no current flow across the wire.

## The magnetic field between two parallel wires carrying current in the same direction

The magnetic field exerts an attractive force when the magnetic field between two parallel wires carries current in the same direction. With the current in the same direction, most of the field is canceled out, but some of the remaining fields tend to pull the wires towards one another, forming an attractive force.

The magnetic force thus generated follows Biot-Savart’s law. According to the law, if the current between the two parallel wires flows in the same direction should attract. For example, consider the current flowing in two parallel wires in towards upward direction. Generally, the magnetic field lines travel from the north to the south pole. These field lines normally flow from left to right perpendicular to the wire.

When the current flowing in two parallel wires is in the same direction, the magnetic field fields in the wire are created so that the south pole of one wire faces the north pole of the other wire. Thus there will be an attraction between both wires as we know opposite poles attract each other. When the magnetic field between the two parallel wires carries the current in the same wire, it acts as an elastic band, which tends to shorten as much as possible.

## The magnetic field between two parallel wires carrying current in the opposite direction

A repulsive force will be created when the magnetic field between two parallel wires carries the current in the opposite direction. The wires repel so that there is a limit for shortening the fields.

According to Biot-Savart’s law, the current in the opposite direction in two parallel wires must repel because when current flows in the opposite direction to one another, the magnetic field generated by the current strictly follows the right-hand rule. When the current flows in the opposite direction, the magnetic field is created in the wire so that one wire’s north pole faces the other wire’s north pole. Thus, repulsive force is exerted.

The current flowing in the opposite direction acts as current in a series circuit. The magnetic field produced by the current moves in the same direction at the point they intersect. Since like poles always repel, the magnetic field produced due to the current flowing in the parallel wires in the opposite direction repels.

## Solved Problems on the magnetic field between two parallel wires

### Calculate the total magnetic field between two parallel wires of length 5m. The current flowing in wire 1 is 2.5amps, and wire 2 is 1.67amps. The distance between the wire is 1m, the point at which the magnetic field is observed is 4m, the wire is 2m, and the point of observation is 3.6m. The current flowing in both wires is in the same direction. And also calculate the magnetic force exerted between the parallel wires.

Solution:

Given –current flowing in the wire 1 I1=2.5amps.

The current flowing in the wire 2 I2=1.67amps.

The distance between wire1 and the point of observation r1=4m.

The distance between wire2 and the point of observation is r2=3.6m.

Length of both the wires l=5m.

The total magnetic field between two parallel wires of the unequal distance between the point of observation is given by

Substituting the values, we get

But μ0=4π×10-7NA-2

Substituting the given values and the value of μ0, we get

B=0.322×10-7Tesla.

The force exerted between two parallel wires

Substituting all the values, we get force F=1.043×10-7N.

### Two wires, M and N, of lengths 12m and 16m, respectively, are separated by a distance of 20cm. Both wires carry the current of 23amps and 25amps respectively in opposite directions. Find the position at which the resultant magnetic field becomes zero.

Solution:

Given –the length of the wire M is L1=12cm=0.12m.

The length of the wire N is L2=16cm=0.16m.

The current carried by the wire M is I1=23amps

The current carried by the wire, N is I2=25amps

Let x be the point from wire M, where the magnetic field is zero. Since the current is flowing in the opposite direction between wire M and N; thus, at point x+r away from wire N, the magnetic will be zero.

The magnetic field due to wire M is B1; it is given by

The magnetic field due to wire N is B2, given by the equation

The separation between the wires M and N is r which is mentioned in the problem as r=20cm=0.20m. Substituting the value in the above equation

The resultant magnetic field will be zero when B1=B2, so equating both the equation

On solving, we get

Cross multiplying the above equation, we get

23(x+0.20)=25x

23x+4.6=25x

4.6=25x-23x

4.6=2x

x=2.3

The magnetic field will be zero at the point 2.3m away from the wire M.

### Two wires, A and B, are kept parallel, separated by a distance of 4cm. Both wires carry the current of 12amps and 8amps in the same direction, respectively. Find the point away from wire B where the magnetic field between two parallel wires A and B is zero.

Solution:

Given –the current carried by the wire, A I1=12amps

The current carried by the wire B is I2=8amps

The separation between wire A and B is r=4cm=0.04m

Let x be point away from A, where the magnetic field between A and B will be zero.

Since the current in both wires flows in the same direction, the point from wire B, where the magnetic field between A and B will be zero, is given by (r-x)

The magnetic field at point x due to wire A is

The magnetic field due to wire B is

The magnetic field between two parallel wires will be zero when the magnitude of both wires carrying current is the same.

i.e., BA=BB

Solving and cross multiplying the equation,

8x=12(0.04-x)

8x=0.48-12x

12x+8x=0.48

20x=0.48

x=0.024m.

The magnetic field is zero at the point 0.024m away from wire A.

The point (r-x) gives the point from wire B where the magnetic field is zero.

r-x=0.04-0.024

r-x=0.016m

At the point 0.016m away from point B, the magnetic field between two parallel wires, A and B, is zero.

#### Conclusion

The value of the magnetic field between two parallel wires highly depends on the direction of the current flow across the wire, and the force exerted due to the magnetic field between parallel wires is correlated to the sin of the angle between the current and field.

Keerthi Murthi

I am Keerthi K Murthy, I have completed post graduation in Physics, with the specialization in the field of solid state physics. I have always consider physics as a fundamental subject which is connected to our daily life. Being a science student I enjoy exploring new things in physics. As a writer my goal is to reach the readers with the simplified manner through my articles. Reach me – keerthikmurthy24@gmail.com