Lecture #4.4 Magnetic Field During last several lectures we have been discussing electromagnetic phenomena. However, we only considered examples of electric forces and fields. We first talked about electrostatics which studies electric charges at rest. We then discussed motion of electric charges which is known as electric current. But in both cases our discussion only had to do with electric phenomena. The other part of electromagnetism studies magnetic phenomena. Magnetic interaction is similar to electric interaction in a sense that it is also long-range interaction which occurs between the objects at large distances without actual contact. This means that, once again, we can introduce the concept of field. Now we know that there is a close relationship between electric and magnetic fields. They both demonstrate two different aspects of the same electromagnetic interaction. However, this was not realized until late 19 th century, when James Clerk Maxwell formulated his set of the Maxwell equations. So, what is magnetic field and how it is different and/or similar to electric field? Probably the most well-known example of magnetic interaction is the example of permanent magnets. We use them to stick something to refrigerator s door or as a compass to determine direction of magnetic field of the Earth. Even though a permanent magnet is not the best example to understand the basic properties of magnetic interaction, we can still make several important conclusions based on the properties of permanent magnets. First of all we can see that magnetic force as well as electric force gets weaker with increase of distance between the interacting objects. Moreover, if one performs measurements, it can be shown that force acting between the point-like magnets decreases according to the same law, inversely proportional to the square of the distance. Secondly, we can see that similar to the case of positive and negative charges, each magnet has two poles. Magnets are attracted if they are oriented with different poles towards each other and they are repelled if the same poles are directed towards each other. Magnets can have different strength. Some magnets are stronger than others and the force of their interaction is higher. You can also notice that similar to electric phenomena not all the substances are alike in their magnetic properties. We know that some metals, such as iron or steel, exhibit magnetic properties, while others such as silver or gold as well as majority of nonmetals never become permanently magnetized.
The major difference between electric and magnetic phenomena is that each permanent magnet has two poles. In contrast to electric charges it is impossible to separate positive from negative. Every time when you cut magnet in half, it still has both poles. There are two different types of magnetic poles, but how can we distinguish between them. Each permanent magnet can be used as a compass. If suspended freely one end of the magnet will point in the northern direction and is known as the north pole of the magnet, another end will orient towards the south and is known as the south pole of the magnet. The same poles of different magnets are repelled, different poles are attracted. Magnets are always magnetic dipoles, it is impossible to find the magnet which only has one pole. Since magnetic dipoles are interacting with each other and with the earth, earth itself must be a large magnetic dipole having north magnetic and south magnetic poles. Even though those poles are not exactly at the same locations as geographical north and geographical south poles, but they are rather close. Note that the south magnetic pole is close to the north geographical pole and the north magnetic pole is close to the south geographical pole. So, what are the reasons for magnetic interactions and why are they different from electric interactions not allowing creation of magnetic monopoles? To understand that, we have to find out where else magnetic interactions occur besides between permanent magnets. The simple experiment shows that if a compass needle is placed near a wire carrying electric current it will deflect if originally the wire is oriented in the same direction as magnetic needle. This means two things. The first is that electric current creates magnetic field and the second is that magnetic field of the current is probably oriented perpendicular to original magnetic field of the needle, causing it to change its orientation. One can introduce the concept of magnetic field lines in a same way as we have introduced the concept of electric field lines. These lines are tangential to the direction of the magnetic field at every point in space and the number of lines per unit of area is proportional to the strength of the magnetic field B. Assuming that magnetic needle is always oriented in such a way that its own magnetic field is in the same direction as external magnetic field, we can reproduce the picture of magnetic field created by a permanent magnet of by electric wire if we use the small iron filings sprinkled onto smooth surface. Those filings will align in the direction of the field. They are bunched together near the poles of the permanent magnet, showing that magnetic field is stronger closer to the poles of the magnet. We may also see that magnetic field lines are starting at one of the poles of the magnet and ending at the other pole (similar to situation of electric dipole). So we can define direction of magnetic field in a following way. The magnetic field lines are staring at the north pole of the magnet and ending at the south pole of the magnet, so the magnetic field is directed from the north pole to the south pole. Since
there are no magnetic monopoles, magnetic field lines are all closed loops. They would even continue inside of magnets, not having any starting or ending points. This is different compared to electric field lines. Even though the real magnetic field of the earth is very complex in its details, but in the simplest approximation we can consider it as a magnetic field of the huge bar magnet with magnetic field lines going from the south geographical pole towards northern geographical pole. We have already seen that electric current produces its own magnetic field, since magnetic needle is deflected in the presence of a wire caring the current. Electric current is the flow of electric charges. This means that magnetic field has to do with moving electric charges. Similarly the external magnetic field should effect moving electric charge. Experiment confirms that this interaction is indeed taking place. Recall that we have introduced electric field as the electric force acting on unit positive testing charge. The similar situation occurs for magnetic force. The magnitude of this force is proportional to magnetic field as well as electric charge. But since magnetic interaction only takes place for moving charges, this force is also proportional to the speed of the charge. The force also depends on the direction of motion of the charge relative to the direction of external magnetic field. In fact, the only component of velocity which affects this force is the component of velocity, v, perpendicular to the magnetic field. The resultant force it is always directed perpendicular to the charge s velocity. So we can say that force acting on the moving charge in the external magnetic field B has the magnitude of F q v B (4.4.1) This, on the other hand, gives us a definition of magnetic field similar to the one we have introduced for electric field. The strength of magnetic filed is F B (4.4.2) qv the force acting per unit of charge per unit of velocity v in the direction perpendicular to the force. 1N The unit of magnetic filed has a special name Tesla 1T. 1C 1m s The only component of velocity affecting magnetic force is the component of velocity in the direction perpendicular to the field. So, if there is an angle between the direction of the field and the velocity, we can say that force has the magnitude F q vbsin (4.4.3) In order to find the direction of this magnetic force one should use the right-hand rule: Point the fingers of your right hand in the direction of the charge s velocity and put your arm so direction of the
magnetic field lines is the direction out of your palm then your thumb will show the direction of the magnetic force acting on the positive charge. Let us now see how external magnetic field affects the motion of charged particles. We have already seen that magnetic force is always perpendicular to the particle s velocity. According to the Newton s second law this force provides acceleration. This acceleration causes the particle to change the direction of its motion by making it turn. If a particle moves in a uniform magnetic field in the plane perpendicular to that external magnetic field, then this field will always remain perpendicular to the particle s velocity. As a result, the same (in absolute value) magnetic force acts on the particle all the time, providing acceleration perpendicular to the particle s velocity at any point of its trajectory. The only way for this to be possible is if the particle is moving around the circle and acceleration is the centripetal acceleration, so F qvb, ma qvb, 2 v m qvb, R v q B R m It is easy to see that the ratio between the speed and the radius is proportional to the charge/mass ratio for the particle. So, if one can measure the speed and the radius of the trajectory for a particle moving in a given magnetic field, then q/m ratio for that particle can be determined. The original speed of the charged particle entering magnetic field can be provided by accelerating this particle through a given potential difference and it is easy to control by adjusting this potential difference. If several particles of different masses and charges are moving with the same speed and enter the same magnetic field they will travel around the circles of different radii. This is how a mass-spectrometer works, allowing sorting particles of different masses according the radii of their paths in magnetic field. In general, a particle may have the velocity which is not necessarily perpendicular to the external magnetic field. If this happens then the component of the particle s velocity, which is perpendicular to the field, will be responsible for the circular motion of the particle. On the other hand the component of the particle s velocity, which is parallel to the magnetic field, will not be affected by this field at all. So, the particle will continue translational motion with constant speed along that direction. The resultant motion of the particle is combination of the two: circular motion around the given direction and translational motion along the same direction which is helical motion.
Example 4.4.1 An electron is accelerated from rest through a potential difference of 310V enters the region of constant magnetic field. If the electron follows the circular path with radius of 17cm, what is the magnitude of this magnetic field? Let us now see, what happens when a particle moves in the region of space where both electric and magnetic fields are present. In this situation the net force acting on the particle is a vector sum of magnetic force and electric force FE qe (4.4.4)
The direction of magnetic force is perpendicular to both the direction of particle s velocity and the direction of magnetic field. The direction of electric force is the same (for positively charge particle) as the direction of the electric field. Let us have positively charged particle, which moves in the direction of axis X and affected by external magnetic field directed in positive Y direction and external electric field directed in the negative Z direction. According to the right hand rule, the direction of magnetic force in this case is in the positive Z-direction. So, by adjusting both or one of the electric or magnetic fields, we can reach a situation, where the total force acting on the particle is zero. This means that particle has no acceleration and will continue moving in positive X-direction with the same speed as if nothing has happened. There is a certain relation between the speed of the particle, electric and magnetic fields in this case. Indeed we have F F E 0, qvb sin 90 qe 0, vb E 0, E v B o So, if the force acting on the particle is zero, it can only have speed which obeys this equation. Since external magnetic field affects motion of the charged particles, it should also affect the current-carrying wire, because electric current consists of moving electric charges. We saw how electric charges inside of the wire were shifted by external magnetic field. Positive charges are shifted to one side of the wire and negative charges are shifted to the other side of the wire. As a result the potential difference is produced between the opposite sides of this wire. This is called the Hall s effect. Now let us determine the force acting on a current-carrying wire in the presence of external magnetic field. This force depends on how long the wire is, since the longer wire has more charged particles in it, and the force is larger. So, let L be the length of the wire. We can introduce it as a vector which has the same direction as the direction of the current in this wire. Let q be the charge passing through this wire of length L during time interval t. This means that the velocity of charges inside of L the wire is v. All the charges are moving in the same direction and so the absolute value of the t total force acting on these charges is where I is the current in the wire. L F qvb sin q Bsin ILB sin t, (4.4.5)
If the angle between the direction of the current and external magnetic field is 90 degrees then the equation becomes extremely simple F IlB (4.4.6) The direction of this force obeys the same right-hand rule as the force acting on moving charge. In order to have the electric current you have to have a closed circuit. So, every time when we are talking about electric currents, we are rather talking about closed loops than straight wires. That is why, we have to know what is magnetic field produced by a loop and how the loop of current is affected by the external magnetic field. Knowing that magnetic field lines are the circles surrounding the wire, it is easy to see that magnetic field should be the strongest at the center of the loop and it should be in the direction perpendicular to the surface of the loop which is essentially the same as magnetic field of the bar magnet. This, by the way, explains existence of the permanent magnets and absence of magnet monopoles. Indeed, the magnetic field of any permanent magnet is due to the presence of many micro-currents inside of this magnet. If all the currents are oriented in a same way (which happens during magnetization process, when the sample is placed in the strong external magnetic field) then the net field of all these micro-currents results in the magnetic field of this permanent magnet. So, as we can see, if the loop of wire carrying electric current is placed in the external magnetic field, it will try to change its orientation in a same way as a magnetic needle does near the permanent magnet. It will try to orient in such a way that the magnetic field produced by the current of the loop is directed along with the external magnetic field. This means that it will be the net torque acting on the loop. That happens because of the magnetic forces acting on both sides of the loop in opposite directions, causing the loop to rotate. Most part of electric motors works based on this principle. However, in order to keep motor working, the orientation of the coil should change all the time, which can be achieved by changing the direction of the current in the coil every half of the turn. If one has a rectangular loop, which is able to rotate around the axis of rotation passing through the center points of the two sides of this rectangle, then it will be magnetic force acting on two other sides of the loop. Those forces are acting in opposite directions, since current goes in opposite directions for the opposite sides of the loop. Each of the forces can be calculated according to equation 4.4.6. If the sides of this rectangle have lengths a and b, the axis of rotation passes through the centers of the sides with length b, and magnetic field is in the surface of the loop, then the absolute value of the torque produced by each of the forces has its maximum value of 1 2 biab
The net torque due to the forces on both sides is where A=ab is the area of the loop. 1 2 biab abib AIB, (4.4.7) 2 If direction of magnetic field is not in the surface of the loop then forces acting on the loop are not perpendicular to the surface of the loop. That means that the torque is also dependent on the angle between the direction of external magnetic field and the normal to the surface of the loop. The net torque is AIBsin (4.4.8) Even though we derived this equation for rectangular loop, it has very general nature and works for loops of any shape. Also if a loop consists of not just one but several turns of electric wires then the torque in the equation 4.4.8 will be larger by the number of turns.