Magnetic Field Magnetic Force In our study of electricity, we described the interactions between charged objects in terms of electric fields. Recall that an electric field surrounds any electric charge. In addition to containing an electric field, the region of space surrounding any moving electric charge also contains a magnetic field. A magnetic field also surrounds a magnetic substance making up a permanent magnet. Historically, the symbol B has been used to represent a magnetic field, and this is the notation we use in this text. The direction of the magnetic field B at any location is the direction in which a compass needle points at that location. As with the electric field, we can represent the magnetic field by means of drawings with magnetic field lines. Fig.1 Figure.1 shows how the magnetic field lines of a bar magnet can be traced with the aid of a compass. Note that the magnetic field lines outside the magnet point away from north poles and toward south poles.
The Magnetic Field of the Earth The Earth s south magnetic pole is located near the north geographic pole, and the Earth s north magnetic pole is located near the south geographic pole. Magnetic Force Figure.2 We can define a magnetic field B at some point in space in terms of the magnetic force F B that the field exerts on a charged particle moving with a velocity v, which we call the test object. For the time being, let us assume that no electric or gravitational fields are present at the location of the test object. Experiments on various charged particles moving in a magnetic field give the following results: The magnitude F B 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 F B 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. When the particle s velocity vector makes any angle θ 0 with the magnetic field, the magnetic force acts in a direction perpendicular to both v and B; that is, F B is perpendicular to the plane formed by v and B (Fig. 3-a). The magnetic force exerted on a positive charge is in the direction opposite the direction of the magnetic force exerted on a negative charge moving in the same direction (Fig. 3-b). The magnitude of the magnetic force exerted on the moving particle is proportional to sin θ, where θ is the angle the particle s velocity vector makes with the direction of B. We can summarize these observations by writing the magnetic force in the form Fig.3
Right-hand rules for determining the direction of the magnetic force The vector v is in the direction of your thumb and B in the direction of your fingers. The force F B on a positive charge is in the direction of your palm. The force on a negative charge is in the opposite direction. Figure 4 Notation used to indicate the direction of B Figure 5 (a) Magnetic field lines coming out of the paper are indicated by dots, representing the tips of arrows coming outward. (b) Magnetic field lines going into the paper are indicated by crosses, representing the feathers of arrows going inward.
In the SI unit of magnetic field is the newton per coulomb-meter per second, which is called the tesla (T): or Example1: An electron in a television picture tube moves toward the front of the tube with a speed of 8.0 x 10 6 m/s along the x axis (Fig. 6). Surrounding the neck of the tube are coils of wire that create a magnetic field of magnitude 0.025 T, directed at an angle of 60 to the x axis and lying in the xy plane. (A) Calculate the magnetic force on the electron (B) Find a vector expression for the magnetic force on the electron (A) Fig. 6 (B)
Fig. 7 Magnetic Force Acting on a Current-Carrying Conductor If a magnetic force is exerted on a single charged particle when the particle moves through a magnetic field, it should not surprise you that a current-carrying wire also
experiences a force when placed in a magnetic field. This follows from the fact that the current is a collection of many charged particles in motion; hence, the resultant force exerted by the field on the wire is the vector sum of the individual forces exerted on all the charged particles making up the current. To demonstrate the magnetic force acting on a current-carrying conductor we hang a wire between the poles of a magnet. The magnetic field is directed into the page. When the current in the wire is zero, the wire remains vertical. However, when the wire carries a current directed upward, the wire deflects to the left. If we reverse the current, the wire deflects to the right. Figure.8 The magnetic force on the wire is given by: Where L is a vector that points in the direction of the current I and has a magnitude equal to the length L of the segment. Note that this expression applies only to a straight segment of wire in a uniform magnetic field.
Now consider an arbitrarily shaped wire segment of uniform cross section in a magnetic field. The magnetic force exerted on a small segment of vector length ds in the presence of a field B is The total force F B acting on the wire is: Particular cases 1- If B is uniform so, The quantity which represents the vector sum of all the length elements from a to b is equal to the vector L directed from a to b. Therefore, Conclusion: The magnetic force on a curved current-carrying wire in a uniform magnetic field is equal to that on a straight wire connecting the end points and carrying the same current. 2- An arbitrarily shaped closed loop carrying a current I is placed in a uniform magnetic field. Since,
we conclude that F B = 0; that is, the net magnetic force acting on any closed current loop in a uniform magnetic field is zero. Quiz The four wires shown in the figure below all carry the same current from point A to point B through the same magnetic field. In all four parts of the figure, the points A and B are 10 cm apart. Rank the wires according to the magnitude of the magnetic force exerted on them, from greatest to least. Figure.9 Example2 A wire bent into a semicircle of radius R forms a closed circuit and carries a current I. The wire lies in the xy plane, and a uniform magnetic field is directed along the positive y axis, as shown in the figure below. a- Find the magnitude and direction of the magnetic force acting on the straight portion of the wire and on the curved portion. b- Deduce the net force exerted on the loop.
Figure.10 Solution a- The magnetic force F 1 acting on the straight portion has a magnitude F 1 = ILB = 2IRB because L= 2R and the direction of F 1 is out of the page based on the righthand rule for the cross product L ^ B. The magnetic force on the curved portion is the same as that on a straight wire of length 2R carrying current I to the left. Thus, F 2 = ILB = 2IRB. The direction of F 2 is into the page based on the right-hand rule for the cross product L ^ B. b- Motion of a Charged Particle in a Uniform Magnetic Field The magnetic force acting on a charged particle moving in a magnetic field is perpendicular to the velocity of the particle and that consequently the work done by the magnetic force on the particle is zero. Now consider the special case of a positively charged particle moving in a uniform magnetic field with the initial velocity vector of the
particle perpendicular to the field. Let us assume that the direction of the magnetic field is into the page, as in Figure 4. As the particle changes the direction of its velocity in response to the magnetic force, the magnetic force remains perpendicular to the velocity. If the force is always perpendicular to the velocity, the path of the particle is a circle! Figure 4 shows the particle moving in a circle in a plane perpendicular to the magnetic field. The particle moves in a circle because the magnetic force F B is perpendicular to v and B and has a constant magnitude q v B. As Figure 4 illustrates, the rotation is counterclockwise for a positive charge. If q were negative, the rotation would be clockwise. We can equate the magnetic force to the product of the particle mass and the centripetal acceleration as the following : That is, the radius of the path is proportional to the linear momentum m v of the particle and inversely proportional to the magnitude of the charge on the particle and to the magnitude of the magnetic field. The angular speed of the particle is The period of the motion (the time interval the particle requires to complete one revolution) is equal to the circumference of the circle divided by the linear speed of the particle:
Example 3 A proton is moving in a circular orbit of radius 14 cm in a uniform 0.35-T magnetic field perpendicular to the velocity of the proton. Find the linear speed of the proton. References This lecture is a part of chapter 27 from the following book Physics for Scientists and Engineers (with Physics NOW and InfoTrac), Raymond A. Serway - Emeritus, James Madison University, Thomson Brooks/Cole 2004, 6th Edition, 1296 pages.