Relation between current and drift velocity (Section 18.3)
Summary
Magnetic field lines are interpreted just like electric field lines. The magnetic field at any point is tangent to the field line; the magnitude of the field is proportional to the number of lines per unit area perpendicular to the lines.
Magnetic field lines are always closed loops because there are no magnetic monopoles.
The smallest unit of magnetism is the magnetic dipole. Field lines emerge from the north pole and reenter at the south pole. A magnet can have more than two poles, but it must have at least one north pole and at least one south pole.
The magnetic force on a charged particle is
FB = qv × B
(19-5)
If the charge is at rest (v = 0) or if its motion is parallel or antiparallel to the magnetic field <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif:: ::/sites/dl/free/0070524076/58002/image19_5_text.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (1.0K)</a> then the magnetic force is zero. The force is always perpendicular to the magnetic field and to the velocity of the particle.
The magnitude of the cross product of two vectors is the magnitude of one vector times the perpendicular component of the other:
The direction of the cross product is the direction perpendicular to both vectors that is chosen using right-hand rule 1:
Right-hand rule 1 for cross products (see Fig. 19.9)
Point the thumb of your right hand in a direction perpendicular to both vectors and the outstretched four fingers of your right hand along the first vector. Curl your outstretched fingers inward toward your palm until your fingertips point in the direction of the second vector. If the fingers sweep through an angle less than 180°, the thumb points in the correct direction of the cross product.
If a charged particle moves at right angles to a uniform magnetic field, then its trajectory is a circle. If the velocity has a component parallel to the field as well as a component perpendicular to the field, then its trajectory is a helix.
The magnetic force on a straight wire carrying current I is
F = IL × B
(19-12a)
where L is a vector whose magnitude is the length of the wire and whose direction is along the wire in the direction of the current.
The magnetic torque on a planar current loop is
τ = NIAB sin θ
(19-13b)
where θ is the angle between the magnetic field and the dipole moment vector of the loop. The direction of the dipole moment is perpendicular to the loop as chosen using right hand rule #1 (take the cross product of L for any side with L for the next side, going around in the same direction as the current).
The magnetic field at a distance r from a long straight wire has magnitude
The field lines are circles around the wire with the direction given by right-hand rule 2:
Right-hand rule 2 for the field due to a current (see Fig. 19.30d)
Point the thumb of the right hand in the direction of the current in the wire. Then curl the fingers inward toward the palm; the direction that the fingers curl is the direction of the magnetic field lines around the wire.
The magnetic properties of ferromagnetic materials are due to an interaction that keeps the magnetic dipoles aligned within regions called domains, even in the absence of an external magnetic field.
