Magnetic fields The symbol we use for a magnetic field is B. The unit is the tesla (T). The Earth s magnetic field is about 5 x 10-5 T. Which pole of a magnet attracts the north pole of a compass? Which way does a compass point on the Earth? What kind of magnetic pole is near the Earth s geographic north pole? What are some similarities between electric and magnetic fields? What are some differences?
Similarities between electric and magnetic fields Electric fields are produced by two kinds of charges, positive and negative. Magnetic fields are associated with two magnetic poles, north and south, although they are also produced by charges (but moving charges). Like poles repel; unlike poles attract. Electric field points in the direction of the force experienced by a positive charge. Magnetic field points in the direction of the force experienced by a north pole.
Differences between electric and magnetic fields Positive and negative charges can exist separately. North and south poles always come together. Single magnetic poles, known as magnetic monopoles, have been proposed theoretically, but a magnetic monopole has never been observed. Electric field lines have definite starting and ending points. Magnetic field lines are continuous loops. Outside a magnet the field is directed from the north pole to the south pole. Inside a magnet the field runs from south to north.
Observing a charge in a magnetic field The force exerted on a charge in an electric field is given by v F = v qe Is there an equivalent equation for the force exerted on a charge in a magnetic field?
Observing a charge in a magnetic field Case 1: The charge is positive, and initially stationary in the field. Electric Field Magnetic Field
Observing a charge in a magnetic field Case 2: The charge is positive, and given an initial velocity parallel to the field. Electric Field Magnetic Field
Observing a charge in a magnetic field Conclusions: Magnetic fields exert no force on stationary charges, or on charges with velocities directed parallel to the field.
Observing a charge in an electric field Case 3: Three objects, one +, one -, and one neutral, have an initial velocity perpendicular to the field. The field is directed down. Positive charge Neutral Negative charge
Observing a charge in a magnetic field Case 3: Three objects, one +, one -, and one neutral, have an initial velocity perpendicular to the field. The field is directed out of the screen. Positive charge Neutral Negative charge
Observing a charge in a magnetic field Magnetic fields exert no force on neutral particles. The force exerted on a + charge is opposite to that exerted on a charge. The force on a charged particle is perpendicular to the velocity and the field. In this special case where the velocity and field are perpendicular to one another, we get uniform circular motion.
Observing a charge in a magnetic field Case 4: The same as case 3, except the magnetic field is doubled. B 2B
Observing a charge in a magnetic field Case 4: The same as case 3, except the magnetic field is doubled. We observe the radius of the path to be half as large. F = 2 mv r Thus, doubling the magnetic field doubles the force - the force is proportional to the magnetic field.
Observing a charge in a magnetic field Case 5: Three positive charges +q, +2q, and +3q are initially moving perpendicular to the field with the same velocity. q 2q 3q
Observing a charge in a magnetic field Case 5: Three positive charges +q, +2q, and +3q are initially moving perpendicular to the field with the same velocity. We observe the radius of the path to vary inversely with the charge. 2 mv Thus, doubling the charge doubles the force, and tripling the charge triples the force - the force is proportional to the charge. F = r
Observing a charge in a magnetic field Case 6: Three identical charges, are initially moving perpendicular to the field with initial velocities of v, 2v, and 3v, respectively. v 2v 3v r = 2 mv F
Observing a charge in a magnetic field Case 6: Three identical charges, are initially moving perpendicular to the field with initial velocities of v, 2v, and 3v, respectively. We observe the radius of the path to be proportional to the speed. However: r = 2 mv F What does this tell us about how the force depends on speed?
Observing a charge in a magnetic field Case 6: Three identical charges, are initially moving perpendicular to the field with initial velocities of v, 2v, and 3v, respectively. We observe the radius of the path to be proportional to the speed. However: r = 2 mv F What does this tell us about how the force depends on speed? The force is proportional to the speed.
Summarizing the observations There is no force applied on a stationary charge by a magnetic field, or on a charge moving parallel to the field. Reversing the sign of the charge reverses the direction of the force. The force is proportional to q (charge), to B (field), and to v (speed).
Summarizing the observations There is no force applied on a stationary charge by a magnetic field, or on a charge moving parallel to the field. Reversing the sign of the charge reverses the direction of the force. The force is proportional to q (charge), to B (field), and to v (speed). The magnitude of the force is F = q v B sin(θ), where θ is the angle between the velocity vector v and the magnetic field B. The direction of the force, which is perpendicular to both v and B, is given by the right-hand rule.
Something to keep in mind A force perpendicular to the velocity, such as the magnetic force, can not change an object s speed (or the kinetic energy). All it can do is make the object change direction.
The right-hand rule Point the fingers of your right hand in the direction of the velocity. Curl your fingers into the direction of the magnetic field (if v and B are perpendicular, pointing your palm in the direction of the field will orient your hand properly). Stick out your thumb, and your thumb points in the direction of the force experienced by a positive charge. If the charge is negative your right-hand lies to you. In that case, the force is opposite to what your thumb says. Simulation
The right-hand rule
Practice with the right-hand rule In what direction is the force on a positive charge with a velocity to the left in a uniform magnetic field directed down and to the left? 1. up 2. down 3. left 4. right 5. into the screen 6. out of the screen 7. a combination of two of the above 8. the force is zero 9. this case is ambiguous - we can't say for certain
Practice with the right-hand rule v and B define a plane, and the force is perpendicular to that plane. The right-hand rule tells us the force is out of the screen. We use a dot symbol to represent out of the screen (or page), and an x symbol to represent into the screen.
Practice with the right-hand rule, II In what direction is the force on a negative charge, with a velocity down, in a uniform magnetic field directed 1. up 2. down 3. left 4. right out of the screen? 5. into the screen 6. out of the screen 7. a combination of two of the above 8. the force is zero 9. this case is ambiguous - we can't say for certain
Practice with the right-hand rule, II Remember that with a negative charge, your right hand lies to you take the opposite direction.
Practice with the right-hand rule, III In what direction is the force on a positive charge that is initially stationary in a uniform magnetic field directed into the screen? 1. up 2. down 3. left 4. right 5. into the screen 6. out of the screen 7. a combination of two of the above 8. the force is zero 9. this case is ambiguous - we can't say for certain
Practice with the right-hand rule, III Magnetic fields exert no force on stationary charges.
Practice with the right-hand rule, IV In what direction is the force on a negative charge with a velocity to the left in a uniform electric field directed out of the screen? 1. up 2. down 3. left 4. right 5. into the screen 6. out of the screen 7. a combination of two of the above 8. the force is zero 9. this case is ambiguous - we can't say for certain
Practice with the right-hand rule, IV We don t v need v the right-hand rule for an electric field, we need F = qe. The force is opposite to the field, for a negative charge.
Practice with the right-hand rule, V In what direction is the velocity of a positive charge if it feels a force directed into the screen from a magnetic 1. up 2. down 3. left 4. right field directed right? 5. into the screen 6. out of the screen 7. a combination of two of the above 8. the force is zero 9. this case is ambiguous - we can't say for certain
Practice with the right-hand rule, V This is ambiguous. The right-hand rule tells us about the component of the velocity that is perpendicular to the field, but it can t tell us anything about a component parallel to the field that component is unaffected by the field.
Charges moving perpendicular to the field The force exerted on a charge moving in a magnetic field is always perpendicular to both the velocity and the field. If v is perpendicular to B, the charge follows a circular path. F = qvb= 2 mv r The radius of the circular path is: r = mv qb
Charges moving perpendicular to the field The radius of the circular path is: r = mv qb The time for the object to go once around the circle (the period, T) is: T 2π r 2 πmv/ qb 2πm = = = v v qb Interestingly, the time is independent of the speed. The faster the speed, the larger the radius, but the period is unchanged.
Circular paths Three charged objects with the same mass and the same magnitude charge have initial velocities directed right. Here are the trails they follow through a region of uniform magnetic field. Rank the objects based on their speeds. 1. 1 > 2 > 3 2. 2 > 1 > 3 3. 3 > 2 > 1 4. 3 > 1 > 2 5. None of the above
Circular paths The radius of the path is proportional to the speed, so the correct ranking by speed is choice 2, 2 > 1 > 3.
Circular paths, II Three charged objects with the same mass and the same magnitude charge have initial velocities directed right. Rank the objects based on the magnitude of the force they experience as they travel through the magnetic field. 1. 1 > 2 > 3 2. 2 > 1 > 3 3. 3 > 2 > 1 4. 3 > 1 > 2 5. None of the above
Circular paths, II The force is proportional to the speed, so the correct ranking by force is also choice 2, 2 > 1 > 3.
Possible paths of a charge in a magnetic field If the velocity of a charge is parallel to the magnetic field, the charge moves with constant velocity because there's no net force. If the velocity is perpendicular to the magnetic field, the path is circular because the force is always perpendicular to the velocity. What happens when the velocity is not one of these special cases, but has a component parallel to the field and a component perpendicular to the field? The parallel component produces straight-line motion. The perpendicular component produces circular motion. The net motion is a combination of these, a spiral. Simulation
Which way is the field? The charge always spirals around the magnetic field. Assuming the charge in this case is positive, which way does the field point in the simulation? 1. Left 2. Right
Spiraling charges Charges spiral around magnetic field lines. Charged particles near the Earth are trapped by the Earth s magnetic field, spiraling around the Earth s magnetic field down toward the Earth at the magnetic poles. The energy deposited by such particles gives rise to??
Spiraling charges Charges spiral around magnetic field lines. Charged particles near the Earth are trapped by the Earth s magnetic field, spiraling around the Earth s magnetic field down toward the Earth at the magnetic poles. The energy deposited by such particles gives rise to the aurora borealis (northern lights) and the aurora australis (southern lights). The colors are usually dominated by emissions from oxygen atoms. Photos from Wikipedia
A mass spectrometer A mass spectrometer is a device for separating particles based on their mass. There are different types we will investigate one that exploits electric and magnetic fields. Step 1: Accelerate charged particles via an electric field. Step 2: Use an electric field and a magnetic field to select particles of a particular velocity. Step 3: Use a magnetic field to separate particles based on mass.
Simulation Step 1: The Accelerator The simplest way to accelerate ions is to place them between a set of charged parallel plates. The ions are repelled by one plate and attracted to the other. If we cut a hole in the second plate, the ions emerge with a kinetic energy determined by the potential difference between the plates. K = q V
Simulation Step 3: The Mass Separator In the last stage, the ions enter a region of uniform magnetic field B /. The field is perpendicular to the velocity. Everything is the same for the ions except for mass, so the radius of each circular path depends only on mass. 2 mv F = qvb = r = r mv qb
Step 3: The Mass Separator 2 mv F = qvb = r = r mv qb The ions are collected after traveling through half-circles, with the separation s between two ions is equal to the difference in the diameters of their respective circles. s= 2r 2r = 2 1 2( m2 m1) v qb
Magnetic field in the mass separator In what direction is the magnetic field in the mass separator? The paths shown are for positive charges. 1. up 2. down 3. left 4. right 5. into the screen 6. out of the screen
Step 2: The Velocity Selector To ensure that the ions arriving at step 3 have the same velocity, the ions pass through a velocity selector, a region with uniform electric and magnetic fields. v F The electric field comes from a set of parallel plates, and exerts a force of on the ions. The magnetic field is perpendicular to both the ion velocity and the electric field. The magnetic force, FM = qvb, exactly balances the electric force when: Ions with a speed of pass straight through. E = v qe qe = qvb E = vb v = E B
Magnetic field in the velocity selector In what direction is the magnetic field in the velocity selector, if the positive charges pass through undeflected? The electric field is directed down. 1. up 2. down 3. left 4. right 5. into the screen 6. out of the screen
Magnetic field in the velocity selector The right-hand rule tells us that the magnetic field is directed into the screen.
Faster ions in the velocity selector If positive ions are traveling faster than the positive ions that pass undeflected through the velocity selector, what happens to them? They get deflected 1. up 2. down 3. into the screen 4. out of the screen
Faster ions in the velocity selector For ions with a larger speed, the magnetic force exceeds the electric force and those ions are deflected up of the beam. The opposite happens for slower ions, so they are deflected down out of the beam. slow just right fast
Negative ions in the velocity selector If the charges passing through the velocity selector were negative, what (if anything) would have to be changed for the velocity selector to allow particles of just the right speed to pass through undeflected? 1. reverse the direction of the electric field 2. reverse the direction of the magnetic field 3. reverse the direction of one field or the other 4. reverse the directions of both fields 5. none of the above, it would work fine just the way it is
Negative ions in the velocity selector If the charges are negative, both the electric force and the magnetic force reverse direction. The forces still balance, so we don t have to change a thing. positive negative
Simulation A cyclotron A cyclotron is a particle accelerator that is so compact a small one can fit in your pocket. It consists of two D- shaped regions known as dees. In each dee there is a magnetic field perpendicular to the plane of the page. In the gap separating the dees, there is a uniform electric field pointing from one dee to the other. When a charge is released from rest, it is accelerated by the electric field and carried into a dee. The magnetic field in the dee causes the charge to follow a half-circle that carries it back to the gap.
A cyclotron Ernest Lawrence won the 1939 Nobel Prize in Physics for inventing the cyclotron. The cyclotron has many applications, including accelerating ions to high energies for medical treatments. A good example is the Proton Therapy Center at Mass General Hospital (see below). After leaving the cyclotron, the beam is steered using magnetic fields.
Magnetic fields in the dees In what direction is the magnetic field in each of the dees? The path shown is for a positive charge. 1. out of the screen in both dees 2. into the screen in both dees 3. out of the screen in the left dee; into the screen in the right dee 4. into the screen in the left dee; out of the screen in the right dee
Increasing the energy You want to increase the speed of the particles when they emerge from the cyclotron. Which is more effective, increasing the potential difference across the gap or increasing the magnetic field in the dees? 1. increasing the potential difference in the gap 2. increasing the magnetic field in the dees 3. either one, they're equally effective The energy increases by K each time the charge crosses the gap, and stays constant in the dees.