1. Home
  2. University physics volume 2
  3. Unit 2. electricity and magnetism
  4. Sources of magnetic fields
  5. Magnetic force between two

12.3 Magnetic force between two parallel currents  (Page 2/2)

<< Chapter < Page
  University physics volume 2     Page 2 / 2
Chapter >> Page >

The definition of the ampere is based on the force between current-carrying wires. Note that for long, parallel wires separated by 1 meter with each carrying 1 ampere, the force per meter is

F l = ( 4 π × 10 −7 T ⋅ m/A ) ( 1 A ) 2 ( 2 π ) ( 1 m ) = 2 × 10 −7 N/m .

Since μ 0 is exactly 4 π × 10 −7 T ⋅ m/A by definition, and because 1 T = 1 N/ ( A ⋅ m ) , the force per meter is exactly 2 × 10 −7 N/m . This is the basis of the definition of the ampere.

Infinite-length wires are impractical, so in practice, a current balance is constructed with coils of wire separated by a few centimeters. Force is measured to determine current. This also provides us with a method for measuring the coulomb    . We measure the charge that flows for a current of one ampere in one second. That is, 1 C = 1 A ⋅ s . For both the ampere and the coulomb, the method of measuring force between conductors is the most accurate in practice.

Calculating forces on wires

Two wires, both carrying current out of the page, have a current of magnitude 5.0 mA. The first wire is located at (0.0 cm, 3.0 cm) while the other wire is located at (4.0 cm, 0.0 cm) as shown in [link] . What is the magnetic force per unit length of the first wire on the second and the second wire on the first?

Figure shows two current carrying wires. Wires form vertices of a right triangle with legs that are 3 centimeters and 4 centimeters long.
Two current-carrying wires at given locations with currents out of the page.

Strategy

Each wire produces a magnetic field felt by the other wire. The distance along the hypotenuse of the triangle between the wires is the radial distance used in the calculation to determine the force per unit length. Since both wires have currents flowing in the same direction, the direction of the force is toward each other.

Solution

The distance between the wires results from finding the hypotenuse of a triangle:

r = ( 3.0 cm ) 2 + ( 4.0 cm ) 2 = 5.0 cm .

The force per unit length can then be calculated using the known currents in the wires:

F l = ( 4 π × 10 −7 T ⋅ m/A ) ( 5 × 10 −3 A ) 2 ( 2 π ) ( 5 × 10 −2 m) = 1 × 10 −10 N/m .

The force from the first wire pulls the second wire. The angle between the radius and the x -axis is

θ = tan −1 ( 3 cm 4 cm ) = 36.9 ° .

The unit vector for this is calculated by

cos ( 36.9 ∘ ) i ^ − sin ( 36.9 ∘ ) j ^ = 0.8 i ^ − 0.6 j ^ .

Therefore, the force per unit length from wire one on wire 2 is

F → l = ( 1 × 10 −10 N/m) × ( 0.8 i ^ − 0.6 j ^ ) = ( 8 × 10 −11 i ^ − 6 × 10 −11 j ^ ) N/m .

The force per unit length from wire 2 on wire 1 is the negative of the previous answer:

F → l = ( −8 × 10 −11 i ^ + 6 × 10 −11 j ^ ) N/m .

Significance

These wires produced magnetic fields of equal magnitude but opposite directions at each other’s locations. Whether the fields are identical or not, the forces that the wires exert on each other are always equal in magnitude and opposite in direction (Newton’s third law).

Got questions? Get instant answers now!

Check Your Understanding Two wires, both carrying current out of the page, have a current of magnitude 2.0 mA and 3.0 mA, respectively. The first wire is located at (0.0 cm, 5.0 cm) while the other wire is located at (12.0 cm, 0.0 cm). What is the magnitude of the magnetic force per unit length of the first wire on the second and the second wire on the first?

Both have a force per unit length of 9.23 × 10 −12 N/m

Got questions? Get instant answers now!

Summary

  • The force between two parallel currents I 1 and I 2 , separated by a distance r , has a magnitude per unit length given by F l = μ 0 I 1 I 2 2 π r .
  • The force is attractive if the currents are in the same direction, repulsive if they are in opposite directions.

Conceptual questions

Compare and contrast the electric field of an infinite line of charge and the magnetic field of an infinite line of current.

Got questions? Get instant answers now!

Is B → constant in magnitude for points that lie on a magnetic field line?

A magnetic field line gives the direction of the magnetic field at any point in space. The density of magnetic field lines indicates the strength of the magnetic field.

Got questions? Get instant answers now!

Problems

Two long, straight wires are parallel and 25 cm apart. (a) If each wire carries a current of 50 A in the same direction, what is the magnetic force per meter exerted on each wire? (b) Does the force pull the wires together or push them apart? (c) What happens if the currents flow in opposite directions?

Got questions? Get instant answers now!

Two long, straight wires are parallel and 10 cm apart. One carries a current of 2.0 A, the other a current of 5.0 A. (a) If the two currents flow in opposite directions, what is the magnitude and direction of the force per unit length of one wire on the other? (b) What is the magnitude and direction of the force per unit length if the currents flow in the same direction?

a. F / l = 2 × 10 −5 N/m away from the other wire; b. F / l = 2 × 10 −5 N/m toward the other wire

Got questions? Get instant answers now!

Two long, parallel wires are hung by cords of length 5.0 cm, as shown in the accompanying figure. Each wire has a mass per unit length of 30 g/m, and they carry the same current in opposite directions. What is the current if the cords hang at 6.0 ° with respect to the vertical?

Figure shows two parallel wires with current flowing in opposite directions that are hung by cords suspended from hooks.
Got questions? Get instant answers now!

A circuit with current I has two long parallel wire sections that carry current in opposite directions. Find magnetic field at a point P near these wires that is a distance a from one wire and b from the other wire as shown in the figure.

Figure shows two current carrying wires. One carries current out of the page; another carries current into the page. Wires form vertices of a right triangle. Point P is the third vertex and is located at a distance b from one wire and distance a from another wire. Distance b is a leg; distance a is a hypotenuse.

B = μ o I a 2 π b 2 into the page

Got questions? Get instant answers now!

The infinite, straight wire shown in the accompanying figure carries a current I 1 . The rectangular loop, whose long sides are parallel to the wire, carries a current I 2 . What are the magnitude and direction of the force on the rectangular loop due to the magnetic field of the wire?

Figure shows a wire that carries the current I1 and a rectangular loop with long sides that are parallel to the wire and carry a current I2. Distance between the wire and the loop is b. Length of the side of the long side of the loop is a, distance of the short side of the loop is b.
Got questions? Get instant answers now!
<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 2' conversation and receive update notifications?

Ask