CHAPTER 27: MAGNETIC FIELD AND MAGNETIC FORCES

• Magnetic phenomena involve the interaction of moving electric charges • A moving charge (or charges, for an electric current) alters the space around

it, producing a magnetic field. A second moving charge (or current) experiences a magnetic force as a result of moving thru this magnetic field.

• This chapter focuses on how moving charges and currents respond to magnetic fields. Later we will study how moving charges and currents produce magnetic fields.

• A permanent magnet is an object made from a material that is permanently magnetized and thus creates its own persistent magnetic field. Permanent magnets exert forces on each other, as well as on a few particular types of metals (mainly iron, nickel, and cobalt).

• The behavior of permanent magnets is often described in terms magnetic poles. For any bar-shaped permanent magnet (a "bar magnet"), one end is referred to as the north pole (N) and the other as the south pole (S).

• Opposite poles attract one another and like poles repel each other. A non-magnetized material that contains, for example, iron, is attracted towards either pole of a permanent magnet.

• A permanent magnet that is allowed to rotate freely will orient itself such

that its north pole points toward the south pole of the Earth’s magnetic field • The south pole of the Earth’s magnetic field is offset slightly from

geographic north pole (located on the axis of the Earth’s rotation) and so the north pole of a compass needle (bar magnet) points approximately in the direction of geographic north – the angle of deviation from actual geographic north is called magnetic declination. Not to be confused with magnetic inclination: the angle that the Earth’s magnetic field makes with a plane that is tangent to the surface of the Earth. For example, magnetic inclination at the magnetic north pole is 90°

and magnetic inclination at the equator is 0°.

Magnetic Poles vs Electric Charge

• Magnetic poles and electric charges share some similar behavior, but they are not the same thing

• You can isolate a positive charge from a negative charge, but this cannot be done for magnetic poles. Cutting a bar magnet in half produces two weaker yet still complete magnets, each with a north pole and a south pole.

• An isolated charge is called an electric monopole. As far as we know, there

is no analogous quantity for magnetism – the fundamental object is a magnetic dipole. There is no experimental evidence thus far for the existence of a magnetic monopole.

• Nevertheless, as we will see, electric and magnetic interactions are fundamentally related to one another and these two types of interactions are collectively referred to as electromagnetic phenomena

• All electromagnetic interactions are governed by a set of four equations known as Maxwell’s equations

Magnetic Field

• In Chapter 21 we introduced the field model to describe electric interactions: 1) A charge distribution establishes an electric field 𝑬𝑬��⃗ everywhere in space 2) If a point charge q is placed in this field, it will experience an electric

force 𝑭𝑭��⃗ = 𝑞𝑞𝑬𝑬��⃗ • The field model of magnetic interactions has a similar description:

1) A moving charge or a current establishes a magnetic field 𝑩𝑩��⃗ everywhere in space (there will also be an electric field present)

2) Any other moving charge or current in presence of the magnetic field will experience a magnetic force

• This chapter focuses on #2 and we will often ignore the effects of any electric fields that are present. We will learn how magnetic fields are created by moving charges/currents (#1) in the next chapter.

• Like electric field lines, which begin on a positive charge and end on a negative charge, magnetic field lines exit thru the north pole and enter thru the south pole of a permanent magnet

Magnetic Force on a Moving Charge

• Five important properties of the magnetic force on a moving charge: 1) The magnitude of the force is directly proportional to the magnitude of

the charge |𝑞𝑞| 2) The magnitude of the force is directly proportional to the magnitude of

the field �𝑩𝑩��⃗ � that causes the force on the moving charge 3) The magnetic force depends on both the magnitude and direction of the

moving charge’s velocity 𝒗𝒗��⃗ 4) This is the strangest property of the magnetic force – the direction of the

magnetic force is perpendicular to both the velocity of the moving charge 𝒗𝒗��⃗ and the magnetic field 𝑩𝑩��⃗ (another way that this is often stated is the direction of 𝑭𝑭��⃗ is always perpendicular to the plane containing the vectors 𝒗𝒗��⃗ and 𝑩𝑩��⃗ )

5) The magnitude of the force is directly proportional to the component of 𝒗𝒗��⃗ perpendicular to 𝑩𝑩��⃗ (in other words it is proportional to the sin of the angle between them)

• Properties #1-3 and 5 imply that the magnitude of the magnetic force is:

𝐹𝐹 = |𝑞𝑞|𝑣𝑣⊥𝐵𝐵 = |𝑞𝑞|𝑣𝑣𝐵𝐵⊥ = |𝑞𝑞|𝑣𝑣𝐵𝐵 sin𝜙𝜙 • The magnetic force is therefore largest on a charge moving on a line

perpendicular to the field (𝜙𝜙 = 90°) and is zero for a charge moving on a line parallel to the field (𝜙𝜙 = 0° or 𝜙𝜙 = 180°)

• Taking into account property #4, we can express the magnetic force as a cross-product

𝑭𝑭��⃗ = 𝑞𝑞(𝒗𝒗��⃗ × 𝑩𝑩��⃗ )

• Since there are two directions perpendicular to the plane containing 𝒗𝒗��⃗ and 𝑩𝑩��⃗ , we use the right hand rule (RHR) to uniquely specify the direction of 𝒗𝒗��⃗ × 𝑩𝑩��⃗

• As a reminder, to use the RHR you point your fingers of your right hand in the direction of the 1st vector (𝒗𝒗��⃗ ) and curl them towards the 2nd vector (𝑩𝑩��⃗ ). Your thumb then points in the direction of 𝒗𝒗��⃗ × 𝑩𝑩��⃗ . Just as with the electric force, you need to pay attention to the sign of q: if q is positive, then the force on it 𝑭𝑭��⃗ is in the same direction as the vector 𝒗𝒗��⃗ × 𝑩𝑩��⃗ ; if q is negative, then the force on it 𝑭𝑭��⃗ is in the opposite direction of the vector 𝒗𝒗��⃗ × 𝑩𝑩��⃗ .

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• The magnetic field B must have the same units as F/qv: (N∙s)/(C∙m). This combination of units is defined as a tesla (T). 1 T = 1 (N∙s)/(C∙m) = 1 N/(A∙m).

• If there is both an electric and magnetic field present, then the force on a moving charge is 𝑭𝑭��⃗ = 𝑞𝑞�𝑬𝑬��⃗ + 𝒗𝒗��⃗ × 𝑩𝑩��⃗ � **SEE EXAMPLE #1**

Magnetic Field Lines, Magnetic Flux, and Gauss’s Law

• The concept of a field line representation of the magnetic field is the same as how we defined it for the electric field (although the field lines themselves behave differently for a magnetic field)

• Magnetic flux ΦB is defined in the same way as electric flux

Φ𝐵𝐵 = �𝑩𝑩��⃗ ⋅ 𝑑𝑑𝑨𝑨��⃗ = �𝐵𝐵⊥𝑑𝑑𝑑𝑑 = �𝐵𝐵 cos𝜙𝜙𝑑𝑑𝑑𝑑

• For a uniform magnetic field crossing a flat surface, the magnetic flux is:

Φ𝐵𝐵 = 𝑩𝑩��⃗ ⋅ 𝑨𝑨��⃗ = 𝐵𝐵𝑑𝑑 cos𝜙𝜙 • Gauss’s law for electric fields says that the total electric flux thru a closed

surface is proportional to the net electric charge enclosed by the surface • If magnetic charges (monopoles) did exist, then Gauss’s law for magnetic

fields would say the same thing • It turns out that Gauss’s law for magnetic fields says that the total magnetic

flux thru any closed surface is always zero • So, in addition to having no experimental evidence for the existence of

magnetic monopoles, there is also theoretical justification given by the 2nd of Maxwell’s equations we have so far learned, called Gauss’s law for magnetic fields: ∮𝑩𝑩��⃗ ⋅ 𝑑𝑑𝑨𝑨��⃗ = 0

• As a reminder, in defining flux, we defined the direction of the area vector as perpendicular to the gaussian surface and pointing outward from it (not inward)

• NOTE: Electric field lines begin on positive charges and end on negative charges. Magnetic field lines, on the other hand, have no beginning or end – they form closed loops

Motion of Charged Particles in a Magnetic Field

• Unlike most forces, the magnetic force is always perpendicular to the velocity of the charged particle on which it acts. Therefore,

The magnetic force does zero work on a moving charged particle

• This means that if the only force that acts on a charged particle is the magnetic force, then the speed of the particle remains constant (its direction, however, will change). This is true of all magnetic fields, even if they are time-dependent and/or spatially nonuniform.

• The figure above shows a charged particle of mass m moving in a plane

perpendicular to a uniform magnetic field. The magnetic force is always perpendicular to the velocity of the particle and can only change the direction of the particle’s motion, not its speed.

• The resulting motion of q is on a circular path at constant speed – uniform circular motion (recall: the radial acceleration is given by v2/r)

• Applying Newton’s 2nd law to this situation, we have |q|vB = mv2/R and the radius of the circle must be: 𝑅𝑅 = 𝑚𝑚𝑚𝑚

|𝑞𝑞|𝐵𝐵. Physically this makes sense because

the larger |q| and/or B is, the larger the force is, the greater the acceleration, and the tighter the circular path is (smaller R). The larger the momentum mv

is, the more difficult it is to change its straight line path and so R is larger. This is true for a negative charge q as well, except the direction around the circle would be opposite (CW) **SEE EXAMPLE #2**

• This is rotational motion, so using the definition of angular speed (see Chapter 9), 𝜔𝜔 = 𝑚𝑚

𝑅𝑅= 𝑣𝑣 |𝑞𝑞|𝐵𝐵

𝑚𝑚𝑚𝑚= |𝑞𝑞|𝐵𝐵

𝑚𝑚. Thus, the number of revolutions (cycles)

per unit time is the frequency f = ω/2pi, which is independent of the radius of the circle and is called the cyclotron frequency. **SEE EXAMPLE #3**

Velocity Selector

• In certain technological applications you need a beam of charged particles to

all have the same speed (e.g., in electron microscopes, mass spectrometers, etc). A device known as a Wien filter or a velocity selector (shown above) consists of perpendicular electric and magnetic fields.

• Take q to be positive. The magnitude of the electric force on q is qE and points upward; the magnitude of the magnetic force is qvB and points downward. These forces will cancel each other (qE = qvB) if v = E/B, and particles moving at this speed will pass thru undeflected since the net force on them is zero. Adjusting E and/or B allows you to select the speed of the particles you want.

• This works the same way for negatively charged particles except the directions of the electric and magnetic forces are reversed

Charge to Mass Ratio of the Electron

• Equating the general expression for the electromagnetic force on a charged particle to its mass x acceleration (by Newton’s 2nd law) shows that its acceleration is given by 𝒂𝒂��⃗ = 𝑞𝑞

𝑚𝑚(𝑬𝑬��⃗ + 𝒗𝒗��⃗ × 𝑩𝑩��⃗ )

• Thus the charge to mass ratio of a particle is an important dynamical quantity in electromagnetism

• In 1897, JJ Thomson showed that cathode rays were composed of previously unknown negatively charged particles, which he calculated must have bodies much smaller than atoms and a very large value for their charge-to-mass ratio. This particle was the electron and was the first subatomic particle to be discovered.

• Thomson deduced the charge-to-mass ratio using a velocity selector • He accelerated the electrons thru a known potential difference V toward a

region with perpendicular electric and magnetic fields • Conservation of mechanical energy implies that the kinetic energy gained by

a particle equals the potential energy lost: ½mv2 = eV

• So, 𝑣𝑣 = �2𝑒𝑒𝑒𝑒𝑚𝑚

when it reaches the velocity selector. Those particles whose

velocities are equal to the ratio E/B will pass straight thru undeflected. Putting these two together gives an expression for the charge-to-mass ratio:

𝑒𝑒𝑚𝑚 =

𝐸𝐸2

2𝑉𝑉𝐵𝐵2

Mass Spectrometer

• A mass spectrometer (shown in the figure above) is an apparatus that uses a velocity selector in conjunction with a secondary magnetic field to infer the masses of atoms and molecules

• Charged particles are accelerated thru a potential difference and a velocity selector blocks all particles except those with speeds equal to E/B. These particles then enter another region with a different uniform magnetic field 𝑩𝑩��⃗ ′ perpendicular to 𝒗𝒗��⃗ and move in circular paths whose different radii are given by 𝑅𝑅 = 𝑚𝑚𝑚𝑚

𝑞𝑞𝐵𝐵′. If all the particles have the same amount of excess charge, then

this gives a way to infer the masses of the various particles based on their respective radii.

Magnetic Force on a Current-Carrying Conductor

• To derive an expression for the force on a current-carrying conductor,

consider a cylindrical segment of wire with cross-sectional area A and length l as shown above. The current is flowing up the page and a uniform magnetic field points into the page.

• The force on a single moving charge in the current is 𝑭𝑭��⃗ = 𝑞𝑞(𝒗𝒗��⃗ × 𝑩𝑩��⃗ ). The drift velocity is the average speed of any charged particle and is parallel to the current so 𝐹𝐹 = 𝑞𝑞𝑣𝑣𝑑𝑑𝐵𝐵.

• The total force on all the moving charges in the cylinder is the total number of particles (𝑛𝑛𝑑𝑑𝑛𝑛) times the force on each particle (𝑞𝑞𝑣𝑣𝑑𝑑𝐵𝐵)

• In Chapter 25 we defined the current density as 𝐽𝐽 = 𝐼𝐼/𝑑𝑑 = 𝑛𝑛𝑞𝑞𝑣𝑣𝑑𝑑 • The total force is then 𝐹𝐹 = (𝑛𝑛𝑑𝑑𝑛𝑛)(𝑞𝑞𝑣𝑣𝑑𝑑𝐵𝐵) = 𝐼𝐼𝑛𝑛𝐵𝐵 for a magnetic field

perpendicular to the direction of the current • If the magnetic field and current are not perpendicular then it is only the

component of the magnetic field perpendicular to the wire that exerts a force on the wire. In that case: 𝐹𝐹 = 𝐼𝐼𝑛𝑛𝐵𝐵⊥ = 𝐼𝐼𝑛𝑛𝐵𝐵 sin𝜙𝜙 and we can express this using a cross product

• 𝑭𝑭��⃗ = 𝐼𝐼(𝒍𝒍 × 𝑩𝑩��⃗ ) gives the force on a straight segment of wire of length l carrying a current I in the presence of a uniform magnetic field B

• If the conductor is not straight and/or the magnetic field is not uniform (in magnitude or direction), you must integrate to find the total force on the current-carrying conductor

• 𝑭𝑭��⃗ = ∫𝑑𝑑𝑭𝑭��⃗ = 𝐼𝐼 ∫(𝑑𝑑𝒍𝒍 × 𝑩𝑩��⃗ ) – this is a path integral over a path which is defined by the conductor. We have to calculate a path integral for both work �𝑊𝑊 = ∫𝑭𝑭��⃗ ⋅ 𝑑𝑑𝒍𝒍� and for potential difference �∆𝑉𝑉 = −∫𝑬𝑬��⃗ ⋅ 𝑑𝑑𝒍𝒍� if the electric field varies in either magnitude or direction along the path or if the path itself is not a straight line. Except, in those calculations, it was a dot product instead of a cross product (see Chapter 23).

• The same result also holds for moving negative charges. For a current consisting of negative charges moving up the page, the drift velocity flips direction. But because q is negative, the force is still in the same direction as it was for a positive charge. **SEE EXAMPLE #4**

Force and Torque on a Current Loop

• Current-carrying conductors often form closed loops, as in the case of a circuit for example, so it is worth the time to take the results we just obtained and apply them to a current-carrying loop

• We will use a rectangular loop of sides a and b for our analysis • In the figure below, the rectangular loop is oriented such that a line

perpendicular to the loop makes an angle 𝜙𝜙 with a uniform magnetic field 𝑩𝑩��⃗ . A battery (not shown) creates a steady current I in the loop. We will analyze each of the four straight line segments separately.

• First, use the RHR to get the directions of the magnetic force on each of the

four segments – these forces are constant in direction and magnitude along

each of the segments so we can take them as acting at the center of the segments. The directions of these four forces are shown in the figure.

• Next, compute the magnitudes of the forces. Along the two sides of length a, the angle between the wire and the magnetic field is 90° and sin 90° = 1. Along the two sides of length b, the angle between the wire and the magnetic field is (90° – 𝜙𝜙) and sin (90° – 𝜙𝜙) = sin 90°cos 𝜙𝜙 – cos 90°sin 𝜙𝜙 = cos 𝜙𝜙

• The forces along the sides of length a both have magnitude F = IaB sin 90° = IaB. They are in opposite directions and therefore cancel.

• The forces along the sides of length b both have magnitude Fʹ = IbB sin (90° – 𝜙𝜙) = IbB cos 𝜙𝜙. They are in opposite directions and therefore also cancel.

The net force on a current loop in a uniform magnetic field is zero

• The two forces 𝑭𝑭��⃗ ′ and −𝑭𝑭��⃗ ′ lie along the same line and therefore cannot exert a net torque on the loop, no matter what the axis of rotation

• The two forces 𝑭𝑭��⃗ and −𝑭𝑭��⃗ do exert a net torque on the loop which causes it to rotate about the y-axis (looking in the +y direction, the loop rotates clockwise – using the RHR the torques (𝝉𝝉�⃗ ) due to each force both point in the +y direction)

• The lever arm for both forces (the perpendicular distance from the line of action of the force to the axis of rotation) is equal to 𝑏𝑏

2sin𝜙𝜙 and so the

magnitude of the net torque is 𝜏𝜏 = 2 �𝑏𝑏2

sin𝜙𝜙� (𝐹𝐹) = (𝐼𝐼𝐵𝐵𝐼𝐼)(𝑏𝑏 sin𝜙𝜙) =𝐼𝐼𝐵𝐵𝑑𝑑 sin𝜙𝜙, where A = ab is the area enclosed by the loop.

• The product IA is called the magnetic dipole moment µ and is analogous the electric dipole p = qd that was defined in Chapter 21. In terms of the dipole moment, the magnitude of the torque on a current loop can be written as 𝜏𝜏 = 𝜇𝜇𝐵𝐵 sin𝜙𝜙.

• As a vector, 𝝁𝝁��⃗ is perpendicular to the plane of the loop in a direction given by a variation of the RHR – curl your fingers in the direction of the current flow and your thumb points in the direction of 𝝁𝝁��⃗ (this is also the direction of the loop’s area vector 𝑨𝑨��⃗ )

• Now we can express the torque on a current loop in a uniform magnetic field as a cross product: 𝝉𝝉�⃗ = 𝝁𝝁��⃗ × 𝑩𝑩��⃗

• The torque is largest when the dipole moment and the magnetic field are perpendicular. The torque is zero when these two vectors are parallel (𝜙𝜙 = 0°) or antiparallel (𝜙𝜙 = 180°) so these angles by definition are equilibria. Just

as for an electric dipole in a uniform electric field, 𝜙𝜙 = 0° is a stable equilibrium and 𝜙𝜙 = 180° is an unstable equilibrium. Note the similarity in the definitions of the net torque for an electric dipole (𝝉𝝉�⃗ = 𝒑𝒑��⃗ × 𝑬𝑬��⃗ ) and for a magnetic dipole (𝝉𝝉�⃗ = 𝝁𝝁��⃗ × 𝑩𝑩��⃗ ). For both situations the torque always rotates the dipole in a direction of decreasing 𝜙𝜙 – towards the stable equilibrium 𝜙𝜙 = 0°.

• The net force on a current loop in a nonuniform magnetic field is not zero in general. To see this, consider a current loop placed in the field of a bar magnet, show in the figure below.

• Since the magnetic dipole moment is already aligned with the field (the

stable equilibrium configuration), there is no net torque on the loop. However, since the field is not uniform, there is net force. Using the RHR at various points on the loop, you should be able to convince yourself that the total force on the current loop is directed towards the left.

Potential Energy for a Magnetic Dipole

• In Chapter 21, using the result we obtained for the torque on an electric dipole in a uniform electric field (𝝉𝝉�⃗ = 𝒑𝒑��⃗ × 𝑬𝑬��⃗ ), we then derived an expression for the corresponding potential energy: 𝑈𝑈 = −𝒑𝒑��⃗ ⋅ 𝑬𝑬��⃗ = −𝑝𝑝𝐸𝐸 cos𝜙𝜙

• By the same reasoning the potential energy associated with the torque on a magnetic dipole in a uniform magnetic field (𝝉𝝉�⃗ = 𝝁𝝁��⃗ × 𝑩𝑩��⃗ ) is 𝑈𝑈 = −𝝁𝝁��⃗ ⋅ 𝑩𝑩��⃗ =−𝜇𝜇𝐵𝐵 cos𝜙𝜙

• All of these results we have derived began with the assumption that the current loop was shaped like a rectangle but, in fact, they are valid for a current loop of any shape, as long as it lies in a plane **SEE EXAMPLE #5**

• A solenoid is a current distribution where a coil of wire is wound into a tightly packed helix, often around a cylindrical iron core – usually the length of a solenoid is much larger than its diameter

• This configuration is essentially N planar current loops, so for a “solenoid

with N turns,” 𝜇𝜇 = 𝑁𝑁𝐼𝐼𝑑𝑑 and 𝜏𝜏 = 𝑁𝑁𝐼𝐼𝑑𝑑𝐵𝐵 sin𝜙𝜙. The magnetic dipole moment is along the axis of the solenoid and its direction is still determined by the RHR as discussed earlier for a single loop. The effect of the torque is still to align 𝝁𝝁��⃗ with the magnetic field.

• The magnetic field of a solenoid is essentially the same as that of a permanent bar magnet (a compass needle behaves essentially like a small bar magnet).

• Using the RHR on the solenoid above, if you curl your fingers in the

direction that the current flows in the loops, then the magnetic dipole moment will point in the direction of your thumb (in this case 𝝁𝝁��⃗ is directed along the axis of the solenoid, up the page).

• If allowed to rotate freely in the presence of some external magnetic field, this field will exert a torque on the solenoid causing it to rotate such that 𝝁𝝁��⃗ is aligned with the field. The same thing happens with a permanent bar magnet. In both cases this is, at a fundamental level, due to the interaction of moving charges with the external magnetic field.

• In the absence of any external magnetic fields, the magnetic dipole moments of the atoms in a magnetizable material such as iron are randomly oriented and its net dipole moment is zero. When placed in a uniform magnetic field, these dipole moment vectors tend to align with the field giving the metal an overall nonzero dipole moment.

• Consider again the figure shown earlier with a current loop in the presence of the (nonuniform) magnetic field due to a bar magnet. The dipole moment inside the bar magnet points from the south pole towards the north pole. The current loop in the figure is equivalent to a bar magnet with its north pole on the left. The bar magnet has its south pole on the right and as we already argued using the RHR, the force exerted on the loop by the bar magnet is attractive.

• So placing a nonmagnetized piece of iron in the presence of the field of a bar magnet causes the dipole moment vectors associated with its atoms to align with the field giving it a nonzero overall dipole moment parallel to the field

of the bar magnet. Then, as we have argued, there will be an attractive force between them because the field is not uniform.

• Changing the polarity of the magnet flips the direction of the bar magnet’s magnetic field and, thus, the direction of the net dipole moment also flips and the force remains attractive

The Hall Effect

• This was discovered in 1879 by Edwin Hall while working on his doctoral thesis in physics. This effect is used in a wide variety of applications that involve some kind of sensor (as a rotation sensor for anti-lock braking systems; “Hall effect joysticks” to operate cranes, backhoes, etc; a Hall effect thruster is used to propel spacecraft once it is far from any gravitational sources, etc…)

• Consider the situations depicted in the Figures above. In both cases there is a

conducting rectangular slab (oriented in the xz plane) with a uniform external magnetic field perpendicular to the slab (in the +y direction) and a current flowing thru it to the right (+x direction).

• For moving positive charges, this corresponds to a drift velocity, vd, that is also in the +x direction (Figure (b)). For moving negative charges, this corresponds to a drift velocity in the –x direction (Figure (a)).

• As we saw earlier, the magnetic force on the moving charge is the same regardless of whether the charge is positive or negative – the direction of the drift velocity flips but, because the charge changes sign, the direction of the magnetic force remains the same (+z direction in this case)

• In either case the moving charges are driven towards the upper edge of the slab. If the charge carriers are negative, then an excess negative charge builds up there, causing a buildup of excess positive charge on the lower edge. As a result of this separation of charge, an electric field 𝑬𝑬��⃗ (and therefore a potential difference, called the Hall voltage) is created that points upward (+z direction) and increases in magnitude proportional to the increase in the amount of charge accumulated on the edges of the slab. This electric field exerts a force 𝑭𝑭��⃗ = 𝑞𝑞𝑬𝑬��⃗ on each moving charge and, since q is negative, the electric force is in the opposite direction as the magnetic force.

• Once the electric force becomes strong enough to balance the magnetic force, the moving charges are no longer deflected upwards and charge ceases to accumulate on the upper and lower edges of the slab

• If the charge carriers are positive (as in Figure (b)), then positive charge accumulates on the upper edge, the polarity of the potential difference flips, and the direction of the electric field flips. However, the electric force still opposes the magnetic force because q is now positive.

• Once the forces balance we have qEz + qvdBy = 0 so, Ez = –vdBy (the sign of the drift velocity depends on the sign of the moving charge)

• Again we express the current density as Jx = nqvd and use it to eliminate the drift velocity. Finally, we have 𝑛𝑛𝑞𝑞 = − 𝐽𝐽𝑥𝑥𝐵𝐵𝑦𝑦

𝐸𝐸𝑧𝑧 and this result is valid for

positive as well as negative charges. • Using this effect, experiments have confirmed that the moving charges in a

conductor really are negative even though the analysis of a circuit is done the same way regardless of whether the moving charges are assumed to be positive or negative. **SEE EXAMPLE #6**