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Magnetic Force Between Two Parallel Conductors
Parallel wires carrying current produce significant magnetic fields, which in turn produce significant forces on currents.
Learning Objective

Express the magnetic force felt by a pair of wires in a form of an equation
Key Points

The field (B_{1}) that that current (I_{1}) from a wire creates can be calculated as a function of current and wire separation (r):
$B_1=\frac {\mu_0I_1}{2\pi r}$ μ_{0} is a constant. 
$F=IlB \sin \theta$ describes the magnetic force felt by a pair of wires. If they are parallel the equation is simplified as the sine function is 1. 
The force felt between two parallel conductive wires is used to define the ampere—the standard unit of current.
Terms

ampere
A unit of electrical current; the standard base unit in the International System of Units. Abbreviation: amp. Symbol: A.

current
The time rate of flow of electric charge.

magnetic field
A condition in the space around a magnet or electric current in which there is a detectable magnetic force, and where two magnetic poles are present.
Full Text
Parallel wires carrying current produce significant magnetic fields, which in turn produce significant forces on currents. The force felt between the wires is used to define the the standard unit of current, know as an amphere.
In , the field (B_{1}) that I_{1} creates can be calculated as a function of current and wire separation (r):
The field B_{1} exerts a force on the wire containing I_{2}. In the figure, this force is denoted as F_{2}.
The force F_{2} exerts on wire 2 can be calculated as:
Given that the field is uniform along and perpendicular to wire 2, sin θ = sin 90 derees = 1. Thus the force simplifies to: F_{2}=I_{2}lB_{1}
According to Newton's Third Law (F_{1}=F_{2}), the forces on the two wires will be equal in magnitude and opposite in direction, so to simply we can use F instead of F_{2}. Given that wires are often very long, it's often convenient to solve for force per unit length. Rearranging the previous equation and using the definition of B_{1} gives:
If the currents are in the same direction, the force attracts the wires. If the currents are in opposite directions, the force repels the wires.
The force between currentcarrying wires is used as part of the operational definition of the ampere. For parallel wires placed one meter away from one another, each carrying one ampere, the force per meter is:
The final units come from replacing T with 1N/(A×m).
Incidentally, this value is the basis of the operational definition of the ampere. This means that one ampere of current through two infinitely long parallel conductors (separated by one meter in empty space and free of any other magnetic fields) causes a force of 2×10^{7} N/m on each conductor.
Key Term Reference
 Law
 Appears in this related concepts: TwoComponent Forces, Physics and Other Fields, and Models, Theories, and Laws
 conductor
 Appears in this related concepts: Physical Properties and Atomic Size, Semiconductors, and ParallelPlate Capacitor
 equation
 Appears in this related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 force
 Appears in this related concepts: Driven Oscillations and Resonance, Glancing Collisions, and First Condition
 magnitude
 Appears in this related concepts: Roundoff Error, Multiplying Vectors by a Scalar, and Components of a Vector
 parallel
 Appears in this related concepts: Charging a Battery: EMFs in Series and Parallel, Resistors in Parallel, and Combination Circuits
 perpendicular
 Appears in this related concepts: The Cross Product, Tangent Vectors and Normal Vectors, and Normal Forces
Sources
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Cite This Source
Source: Boundless. “Magnetic Force Between Two Parallel Conductors.” Boundless Physics. Boundless, 28 Jan. 2015. Retrieved 26 May. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/magnetism21/magneticfieldsmagneticforcesandconductors159/magneticforcebetweentwoparallelconductors5636223/