# Magnetic Deflection in a Cathode Ray Oscilloscope

Before explaining the magnetic deflection in a cathode ray oscilloscope, we need to recall our concept of the magnetic force acting on a conductor in the magnetic field. A current-carrying conductor always experiences a force inside a magnetic field. The expression of force is

Where ‘F’ is the force acting on the conductor. ‘B’ is the flux density of the magnetic field. ‘I’  is the current flowing through the conductor. ‘L’ is the length of the conductor coming under the influence of the magnetic field.

Let us considered there is n number of electrons drifting through the conductor of length L in t seconds. Hence the charge transferring through the conductor in t seconds is

Hence we can write the expression for current through the conductor as

Now we can write the expression of the force acting on the electrons as

The electrons drift through a conductor per unit time is the drift velocity of the electrons. Here, the electrons drift the length L in t seconds. So, here the drift velocity of the electrons will be

Hence the expression of the force ultimately comes as

### Concept of Force Acting on Electron Beam

Therefore, the force per electron will be

So far we have discussed the movement of electrons in the direction perpendicular to the magnetic field. Also, we have derived the force acting on the electron perpendicular to the direction of the magnetic field. Let us visualize that the uniform magnetic field is directed inwards to the plane in front of our eyes. Also, we imagine that the electron is drifting from left to right in the magnetic field. Therefore, the force acting on the electron is directed straight downwards. That means the force acting on the electron is just perpendicular to the direction of the motion of the electron. Hence the force acting on the electron perpendicular to the motion at every instant causes the electron to move in a circular path.

Let us consider r is the radius of the circular path. Thus, the angular acceleration of the electron is

If m is the mass of one electron then we can write the expression of the force as

So, from the previous expression of force per electron, we can write

Now we can write the angular velocity of the motion of the electron as

So the time period of the circular motion of the electron is

## Expression of Magnetic Deflection in a Cathode Ray Oscilloscope

We use these equations for determining the magnetic deflection in a cathode ray oscilloscope

The above figure shows that the width of the magnetic field is l meter. The screen of the cathode ray oscilloscope is located at L meter away from the center of the magnetic field. After entering the magnetic field the electron follows the circular path of radius r meter. Then after leaving the magnetic field the electron follows a straight line to reach the screen. Suppose, θ is the angle of deflection of the electron.

Practically, this theta is a very small angle. So, from the triangle ABC and OPQ, we get

After putting the expression of r in the above equation, we get,

Now suppose Va is the accelerating voltage applied between anode and cathode in the electron gun system to accelerate the electron before entering in the magnetic field. The kinetic energy of the electron just at the entrance of the magnetic field can be written as

Therefore, we can write,

### Sensitivity of Magnetic Deflection in a Cathode Ray Oscilloscope

The sensitivity (S) of the magnetic deflection in a cathode ray oscilloscope is the ratio of the deflection of the electron beam on the screen to the magnetic flux density. Hence we can write

Therefore the sensitivity of the magnetic deflection is inversely proportional to the square root of the accelerating voltage applied to the electron gun.

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