Electric Charge in a Magnetic Field

Electrons play the leading role many areas of science and technology, as they possess electric charge, enabling them to carry current.

Electric charge, or q, is a physical property describing whether a unit of matter has more protons, making it positively charged, more electrons, making it negatively charged or an equal number of protons and electrons, making it uncharged. This fundamental property describes electromagnetic interactions, where like charges are repelled, and opposite charges are attracted.

J.J. Thomson is credited with the discovery of the electron, where he showed that a cathode ray could be deflected by a magnetic field in an evacuated tube. This led to the conclusion that electrons carry a permanent negative charge, and enabled his calculation of the electrons charge-to-mass ratio.

This video will introduce the concept of the force applied to a charge in a magnetic field, and the calculation of the charge-to-mass ratio of an electrode using a cathode ray tube experiment similar to the one used by J.J. Thomson.

Before learning about the cathode ray tube experiment, let's talk about the effects of magnetic field on an electric charge which form the basis of this experiment. When a moving charge is introduced in a magnetic field, the field exerts a force F on the charge.

This is called the Lorentz force. The magnitude of this force is given by the formula qVB sine theta, where q is the magnitude of the charge, V is the velocity, B is the magnitude of the magnetic field, and theta is the angle between the velocity and the magnetic field.

Thus the Lorentz force is maximum when the angle between the V and B is 90 degrees, and is such cases, the direction of the force exerted on a positive charge is given by the right hand thumb rule, making all the vectors perpendicular to each other. If the charge is negative, then the force acts in the opposite direction. Now let's imagine all the magnetic field lines are going into the plane.

These are conventionally denoted by the crosses inside circles. If a positive charge with a velocity perpendicular to the magnetic field is introduced into the field, then the force exerted on this charge will have a direction perpendicular to the velocity. This force has no effect on the magnitude of the velocity, but it affects the direction, and the resultant velocity vector is between the two perpendicular vectors, which forces the charge to move.

Therefore, the charged particles follow a circular path at a constant speed, with all three vectors, velocity, force, and magnetic field perpendicular to each other at all times. Looking at this diagram, it is apparent the force vector represents the centripetal force.

As per Newton's second law, this force is mass of the charge times centripetal acceleration, which is v squared divided by r, the radius of curvature. Recall this force is also given by the Lorentz force formula. Combining these two equations with the law of conservation of energy applicable to the cathode ray tube experiment, we can derive the equation for the charge-to-mass ratio of an electron.

Note that three pieces of information, the potential difference through which electrons are being accelerated, the strength of the magnetic field, and the radius of the circular path followed by the charged particles are needed for the calculation.

Now let's set up and demonstrate how to set up and conduct this cathode ray tube experiment in a physics lab.

First, become familiar with the experimental apparatus. Locate the coils that generate the magnetic field and the digital ammeter, which enables the measurement of current. Locate the double-pole-double-throw switch, which is used to reverse direction of the current, and therefore reverses the magnetic field.

Supply current to the coils that create the magnetic field using the rotary dial. Next, locate the high-voltage supply, which sets the accelerating voltage and an alternating signal of 6.3 V connected to a filament. Electrons are generated by the filament and accelerated by the accelerating voltage.

Now, turn on the high-voltage power supply to turn on the filament. Note that the light that comes on inside the tube is the glowing filament.

Gradually turn up the high voltage to about 2000 V. The part of the screen inside the tube, which is being hit by the electron beam, should glow blue making the electron beam visible.

Next, adjust the current through the coils, creating a uniform magnetic field. Observe that as the current is adjusted up or down, the path of the beam changes. Adjust the current to pass the beam through a particular (x, y) point on the grid. Record the magnitude of the current required to hit this point.

Reverse the current in order to curve the beam in the opposite direction, and adjust the current until the beam passes through the point (x, negative y): or the mirror image of the original point. Record the magnitude of the current. Repeat for four more accelerating voltages, using the same (x, y) and (x, negative y) points.

Observe that as the accelerating voltage is increased and the electrons travel faster, the beam bends less. Thus the coil current must be higher to reach the same (x, y) point.

Next, repeat the full experiment, while this time keeping accelerating voltage constant, and varying the (x,y) and (x, negative y) locations. Collect five data sets, recording the point coordinates and current magnitude for each point and its mirror image.

The radius, r, of the beam path for each accelerating voltage can be calculated using the Pythagorean theorem.

Average the two currents needed to hit both (x,y) and (x, negative y) points for each accelerating voltage to remove the effect of the Earths magnetic field. Do the same for the varied (x, y) and (x, negative y) couples at the same accelerating voltage. Then use the average current to calculate the strength of the magnetic field, B. In the case of this setup, the magnetic field is equal to 0.00423 times the current.

When varying the accelerating voltage, use the value of the magnetic field, the constant radius, and the corresponding voltage to calculate the magnitude of the charge to mass ratio of an electron. Similarly, when varying the (x, y) locations, use the value of the magnetic field, the constant voltage, and the corresponding radius to calculate the electron's charge-to-mass ratio.

Then calculate the average for both varying accelerating voltage and varying (x and y) locations conditions. These experimentally calculated ratio values compare well to the known charge-to-mass ratio of an electron.

Charged particles, which move in a circular path due to an applied magnetic field, have a wide range of applications in technology.

Mass spectrometers identify unknown components of a sample based on their charge-to-mass ratio. Particles travel in a different radius depending on their charge-to-mass ratio, the accelerating voltage and the applied magnetic field. These parameters enable the separation of various components.

Prior to LCD, LED and plasma screen technology, cathode ray tubes, like the experimental set up used in this video, were the basis for all TV screens and computer monitors.

Common lab equipment still uses cathode ray tube displays, such as basic oscilloscopes. The difference being that the deflection of electrons is done via electrostatic deflection, rather than magnetic deflection.

You've just watched JoVE's introduction to electric charges in a magnetic field. You should now understand how electrons are influenced by magnetic fields, and how to use a magnetic field to determine the charge-to-mass ratio of an electron. Thanks for watching!