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Review Problem 1

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Review Problem 2

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Review Problem 3

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Magnetic Fields

lodestone

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Magnetic Fields and Forces

• Properties of the magnetic force on a charge moving in a magnetic field B– The magnitude FB of the magnetic force exerted on the particle is proportional

to the charge q and to the speed v of the particle. – The magnitude and direction of FB depend on the velocity of the particle and on

the magnitude and direction of the magnetic field B. – When a charged particle moves parallel to the magnetic field vector, the

magnetic force acting on the particle is zero.• Magnitude of the magnetic force on a charged particle moving in a

magnetic field

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• The direction of the magnetic force FB acting on a charged particle moving with a velocity v in the presence of a magnetic field B. The magnetic force is perpendicular to both v and B.

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• Two right-hand rules for determining the direction of the magnetic force FB = q v x B acting on a particle with charge q moving with a velocity v in a magnetic field B.

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• the SI unit of magnetic field is the newton per coulomb-meter per second, which is called the tesla (T):

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Magnetic Force Acting on a

Current-Carrying Conductor

• 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.

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• (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.

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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.

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Sources of the Magnetic Field

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Hans Christian Oersted 1819

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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’.

Permeability of free space

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Quick Quiz

• Consider the current in the length of wire shown in figure. Rank the points A, B, and C, in terms of magnitude of the magnetic field due to the current in the length element shown, from greatest to least.

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example

• Consider a circular wire loop of radius R located in the yz plane and carrying a steady current I, as in figure. Calculate the magnetic field at an axial point P a distance x from the center of the loop.

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• The right-hand rule for determining the direction of the magnetic field surrounding a long, straight wire carrying a current. Note that the magnetic field lines form circles around the wire.

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The Magnetic Force Between

Two Parallel Conductors• Two parallel wires that each carry a steady current exert a magnetic

force on each other. The field B2 due to the current in wire 2 exerts a magnetic force of magnitude F1 = I1 l B2 on wire 1. The force is attractive if the currents are parallel (as shown) and repulsive if the currents are antiparallel.

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• The force between two parallel wires is used to define the ampere as follows:When the magnitude of the force per unit length between two long parallel wires that carry identical currents and are separated by 1 m is 2 x 10-7 N/m, the current in each wire is defined to be 1 A.

• The SI unit of charge, the coulomb, is defined in terms of the ampere:When a conductor carries a steady current of 1 A, the quantity of charge that flows through a cross section of the conductor in 1 s is 1 C.

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Ampère’s Law

• When no current is present in the wire, all compass needles point in the same direction (toward the Earth’s north pole).

• 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.

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• The line integral of B.ds around any closed path equals μ0I, where I is the total steady current passing through any surface bounded by the closed path.

• Quick quiz, rank the magnitudes of ∫B.ds for the closed paths in figure, from least to greatest

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• A long, straight wire of radius R carries a steady current I that is uniformly distributed through the cross section of the wire. Calculate the magnetic field a distance r from the center of the wire in the regions r ≥ R and r < R.

example

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The Magnetic Field of a Solenoid

• A solenoid is a long wire wound in the form of a helix. With this configuration, a reasonably uniform magnetic field can be produced in the space surrounded by the turns of wire—which we shall call the interior of the solenoid—when the solenoid carries a current.

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• 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

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• Cross-sectional view of an ideal solenoid, where the interior magnetic field is uniform and the exterior field is close to zero. Ampère’s law applied to the circular path near the bottom whose plane is perpendicular to the page can be used to show that there is a weak field outside the solenoid. Ampère’s law applied to the rectangular dashed path in the plane of the page can be used to calculate the magnitude of the interior field.

n = N/l is the number of turns per unit length

N is the number of turns in the length l

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Gauss’s Law in Magnetism

• the net magnetic flux through any closed surface is always zero:

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The Magnetic Field of the Earth

• The Earth’s magnetic field lines. Note that a south magnetic pole is near the north geographic pole, and a north magnetic pole is near the south geographic pole.