How Does Current Create Force Between Parallel Conductors?
Previously we have learned about the existence of a magnetic field that is due to a current-carrying conductor and the Biot – Savart’s law.
We again have also learned that an external magnetic field that generally exerts a force which is on a current-carrying conductor and the Lorentz force which is the formula that governs this principle.
Thus, from the two studies that we can say that any two current carrying conductors that when placed near each other will exert a magnetic force that is on each other. In this section, we will learn about this case which is in further detail.
Force Between Two Parallel Current Carrying Conductors
We might not generally expect that the force which is between wires is used to define the ampere. We might also be surprised to learn that this force has to do something with why large circuit breakers burn up when they attempt to interrupt large currents.
The force which is between two long straight conductors and the conductors which are parallel as well and separated by a distance r can be found by applying what we have developed in preceding sections. In the figure, we can see the wires and their currents fields which they generally create and the subsequent forces they exert on one another. Let us now consider the field produced by wire 1 and the force it exerts on wire 2, that is we can call the force F2. The field which is due to I1 is at a distance which is r is given to be
B1=μ0I1/2πr
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Figure 1. (a) which is mentioned above is the magnetic field that is generally produced by a long straight conductor perpendicular to a parallel conductor as indicated by RHR-2. Then in the figure that is (b) a view which is from above of the two wires that are shown in (a) with one magnetic field line which is shown for each wire. Here the term that is RHR-1 shows that the force which is between the parallel conductors is attractive when the currents are in the same direction. A similar analysis is shown in this topic that the force is repulsive which is between currents in opposite directions.
If we have three wires which are the parallel in the same plane as it is shown in Figure 2 which is with currents in the outer two running in opposite directions that is it possible for the middle wire to be repelled by both.
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Figure 2. Which we can see is the three parallel coplanar wires with currents in the outer two in opposite directions.
3. Then if we suppose that the two long straight wires run perpendicular to one another without touching. We can say that does one exerts a net force on the other. If so, then we can ask what is its direction? And does one exert a net torque on the other? If so, what is its direction? We need to Justify our responses by using the right-hand rule.
4. The Use which is of the right-hand rule is to show that the force which is between the two loops in Figure 3 is said to be attractive
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Figure 3. We can notice that the two loops of wire carrying currents can exert forces and torques on one another.
5. Electric field can be shielded by the Faraday cage effect.
Now let us consider the system which is shown in the figure above. Here we can see that we have two parallel current carrying conductors that are separated by a distance denoted by ‘d’, So here we can notice that that conductor 2 experiences the same magnetic field that is at every point which is along its length due to conductor 1. That is we can say that the direction of magnetic force is indicated in the figure and is found using the right-hand thumb rule. The direction of the magnetic field which downwards due to the first conductor.
From Ampere’s law of the circuital we can say that the magnitude of the field due to the first conductor can be given by the following:
Ba=μ0I1/2πd
The force which is on a segment of length denoted by letter L of the conductor 2 due to the conductor 1 can be given as follows:
F21=I2LB1=(μ0I1I2/2πd)L
Similarly, we can calculate the force which is exerted by the conductor that is the 2 on the conductor 1. So we can see that conductor which is in the 1 experiences the same force that is generally due to conductor 2 but the direction which is in the opposite. Thus we can pen it down as follows:
F12 = F21
FAQs on Force Between Two Parallel Current Carrying Conductors
1. What is the formula for the force per unit length between two long, straight, parallel current-carrying conductors?
The force per unit length (F/L) between two parallel conductors is given by the formula: F/L = (μ₀ * I₁ * I₂) / (2πr). In this formula:
μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A).
I₁ and I₂ are the currents flowing through the two conductors.
r is the perpendicular distance between the conductors.
2. When is the force between two parallel current-carrying wires attractive, and when is it repulsive?
The nature of the force depends on the direction of the currents in the wires:
Attractive Force: The force is attractive when the currents in both parallel wires flow in the same direction.
Repulsive Force: The force is repulsive when the currents in the parallel wires flow in opposite directions.
3. How is the direction of the force between two parallel conductors determined using the right-hand rules?
The direction of the force is found using a two-step process involving right-hand rules:
First, use the Right-Hand Thumb Rule to determine the direction of the magnetic field produced by one wire at the location of the other. Point your thumb in the direction of the current, and your curled fingers will indicate the direction of the magnetic field lines.
Next, use Fleming's Left-Hand Rule on the second wire. Align your forefinger with the direction of the magnetic field (from step 1), your middle finger with the direction of the current in the second wire, and your thumb will point in the direction of the force experienced by that wire.
4. How is the standard unit of current, the Ampere, defined using the concept of force between parallel conductors?
As per the CBSE 2025-26 syllabus, the Ampere is defined based on the force between two parallel conductors. One Ampere is the constant current which, if maintained in two straight, parallel conductors of infinite length and negligible circular cross-section, placed one metre apart in a vacuum, would produce a force between them equal to 2 × 10⁻⁷ Newtons per metre of length.
5. What are the key assumptions made when deriving the formula for the force between two parallel conductors?
The derivation of the formula for the force between two parallel conductors relies on a few idealised conditions:
The conductors are assumed to be infinitely long. This is done to neglect the complex magnetic field effects that occur at the ends of the wires.
The conductors are perfectly straight and parallel to each other.
The conductors have a negligible circular cross-section.
The entire system is situated in a vacuum or free space (μ = μ₀).
6. Briefly explain the derivation for the force between two parallel current-carrying conductors.
The derivation involves two main steps. Consider two parallel wires carrying currents I₁ and I₂ separated by a distance r.
First, calculate the magnetic field (B₁) produced by wire 1 at the position of wire 2. Using Ampere's Law for a long straight wire, this is B₁ = (μ₀ * I₁) / (2πr).
Next, find the force (F₂) experienced by a length L of wire 2 due to this magnetic field. The formula for magnetic force on a current-carrying wire is F = I * L * B. Therefore, F₂ = I₂ * L * B₁.
Substituting the expression for B₁ into the force equation gives F₂ = I₂ * L * [(μ₀ * I₁) / (2πr)]. Rearranging for force per unit length gives the final expression: F₂/L = (μ₀ * I₁ * I₂) / (2πr).
7. How does the magnetic force between two current-carrying wires differ from the electrostatic force between two stationary charges?
The key difference lies in their origin and nature. Electrostatic force arises from stationary electric charges and is governed by Coulomb's Law; it can be attractive (unlike charges) or repulsive (like charges). In contrast, magnetic force arises from moving charges (electric currents). Its direction depends not on the sign of the charge, but on the relative direction of the currents, and is determined by vector rules like Fleming's Left-Hand Rule.
8. What happens to the magnitude of the force per unit length between two parallel conductors if the distance between them is doubled and the current in one wire is also doubled?
The force remains unchanged. The formula is F/L = (μ₀ * I₁ * I₂) / (2πr). Let the initial force be F. If the current in one wire is doubled (e.g., I₂ becomes 2I₂) and the distance is doubled (r becomes 2r), the new force F' will be: F' / L = (μ₀ * I₁ * (2I₂)) / (2π(2r)) = (2/2) * [(μ₀ * I₁ * I₂) / (2πr)]. The factors of 2 cancel out, so the new force is equal to the original force.
