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Magnetic Force Acting on a Current-Carrying Conductor

(a) Magnetic field lines coming out of the paper are indicated by dots, representing the tips of arrows coming outward. (b) Magnetic field lines going into the paper are indicated by crosses, representing the feathers of arrows going inward.

A segment of a current-carrying wire in a magnetic field B
A segment of a current-carrying wire in a magnetic field B. The magnetic force exerted on each charge making up the current is q vd x B and the net force on the segment of length L is I L x B.

(a) A wire suspended vertically between the poles of a magnet
(a) A wire suspended vertically between the poles of a magnet. (b) The setup shown in part (a) as seen looking at the south pole of the magnet, so that the magnetic field (blue crosses) is directed into the page. When there is no current in the wire, it remains vertical. (c) When the current is upward, the wire deflects to the left. (d) When the current is downward, the wire deflects to the right

The magnetic force exerted on a small segment of vector length ds in the presence of a field B is
A wire segment of arbitrary shape carrying a current I in a magnetic field B experiences a magnetic force. The magnetic force on any segment ds is I ds x B and is directed out of the page. You should use the right-hand rule to confirm this force direction.

(a) A curved wire carrying a current I in a uniform magnetic field
(a) A curved wire carrying a current I in a uniform magnetic field. The total magnetic force acting on the wire is equivalent to the force on a straight wire of length L( running between the ends of the curved wire. (b) A current-carrying loop of arbitrary shape in a uniform magnetic field. The net magnetic force on the loop is zero.

Conclusion the magnetic force on a curved current-carrying wire in a uniform magnetic field is equal to that on a straight wire connecting the end points and carrying the same current the net magnetic force acting on any closed current loop in a uniform magnetic field is zero

Quick Quiz The four wires shown in Figure, all carry the same current from point A to point B through the same magnetic field. In all four parts of the figure, the points A and B are 10 cm apart. Rank the wires according to the magnitude of the magnetic force exerted on them, from greatest to least.

Motion of a Charged Particle in a Uniform Magnetic Field
When the velocity of a charged particle is perpendicular to a uniform magnetic field, the particle moves in a circular path in a plane perpendicular to B. The magnetic force FB acting on the charge is always directed toward the center of the circle.

These results show that the angular speed of the particle and the period of the circular motion do not depend on the linear speed of the particle or on the radius of the orbit. The angular speed ω is often referred to as the cyclotron frequency because charged particles circulate at this angular frequency in the type of accelerator called a cyclotron.

If a charged particle moves in a uniform magnetic field with its velocity at some arbitrary angle with respect to B, its path is a helix if the field is directed in the x direction, as shown in Figure, there is no component of force in the x direction. As a result, ax =0, and the x component of velocity remains constant. However, the magnetic force q v x B causes the components vy and vz to change in time, and the resulting motion is a helix whose axis is parallel to the magnetic field. The projection of the path onto the yz plane (viewed along the x axis) is a circle

Example A proton is moving in a circular orbit of radius 14 cm in a uniform 0.35-T magnetic field perpendicular to the velocity of the proton. Find the linear speed of the proton Solution

The Biot–Savart Law The magnetic field
dB at a point due to the current I through a length element ds is given by the Biot–Savart law. The direction of the field is out of the page at P and into the page at P%

The Magnetic Force Between Two Parallel Conductors

parallel conductors carrying currents in the same direction
attract each other, and parallel conductors carrying currents in opposite directions repel each other.

magnitude in terms of the force per unit length

Ampère’s Law (a) When no current is present in the wire, all compass needles point in the same direction (toward the Earth’s north pole). (b) When the wire carries a strong current, the compass needles deflect in a direction tangent to the circle, which is the direction of the magnetic field created by the current

Ampère’s law describes the creation of magnetic fields by all continuous current configurations

Example

The Magnetic Field of a Solenoid

(a) Magnetic field lines for a tightly wound solenoid of finite length, carrying a steady current. The field in the interior space is strong and nearly uniform. Note that the field lines resemble those of a bar magnet, meaning that the solenoid effectively has north and south poles. (b) The magnetic field pattern of a bar magnet, displayed with small iron filings on a sheet of paper

Magnetic Flux

Magnetic flux through a plane lying in a magnetic field
Magnetic flux through a plane lying in a magnetic field. (a) The flux through the plane is zero when the magnetic field is parallel to the plane surface. (b) The flux through the plane is a maximum when the magnetic field is perpendicular to the plane.

Example

The Magnetic Field of the Earth
the Earth’s south magnetic pole is located near the north geographic pole, and the Earth’s north magnetic pole is located near the south geographic pole south magnetic pole is near the north geographic pole, and a north magnetic pole is near the south geographic pole.

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