Class 12 > Unit # 18:Magnetic Fields > Charge to Mass Ratio (e/m) of an Electron

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Electron Physics - Determination of Charge to Mass Ratio ($e/m$)

Describe the method for determining the ratio of charge to mass ($e/m$) of an electron. Derive relevant mathematical expression.

Experimental Setup & Working Principle

"The charge-to-mass ratio ($e/m$) of an electron was determined by J.J. Thomson using an apparatus consisting of a highly evacuated pear-shaped glass bulb into which several metal electrodes are sealed."

Figure: Schematic diagram of J.J. Thomson's apparatus for measuring the $e/m$ ratio of an electron.

Electrons are produced by heating a tungsten filament $F$ by passing an electric current through it. The emitted electrons are directed toward the screen by applying a negative potential on a hollow cylinder $C$ (open on both sides) surrounding the filament.

Electrons are then accelerated by applying a positive potential to accelerating discs $A$ and $B$. If $V$ is the total potential difference between the accelerating anode discs $B$ and the filament $F$, the potential energy lost by each electron appears as kinetic energy:

$\text{Gain in Kinetic Energy} = \text{Electrical Potential Energy}$

$\frac{1}{2} m v^2 = V e \implies v^2 = \frac{2Ve}{m}$

Where:

• $e$ = charge of an electron
• $m$ = mass of the electron
• $v$ = velocity of the electron
• $V$ = accelerating potential difference

Magnetic Deflection & Centripetal Force

After acceleration, the electron beam passes through the middle of two horizontal metal plates and strikes a screen coated with Zinc Sulphide ($\text{ZnS}$), producing a spot of light at point $O$.

A magnetic field of induction $B$ (directed perpendicularly into the paper) is produced between the plates by two identical current-carrying coils placed on either side of the tube. The magnetic force on the moving electrons acts as a centripetal force, forcing them into a circular path and shifting the light spot from $O$ to $O'$ on the screen:

$F_{\text{magnetic}} = B e v$

Equating magnetic force to centripetal force ($\frac{m v^2}{r}$):

$B e v = \frac{m v^2}{r} \implies B e = \frac{m v}{r}$

Rearranging for velocity $v$:

$v = \frac{B e r}{m} \quad \text{--- (1)}$

Velocity Selection Using Crossed Electric and Magnetic Fields

To determine the exact velocity without relying on $V$ alone, an electric field $E$ is simultaneously established between the plates by applying a potential difference $V_1$, exerting an electrostatic force $E e$ on the electrons opposite to the magnetic force. The potential $V_1$ is adjusted until the two forces neutralize each other, bringing the light spot back to its original unshifted position $O$:

$F_{\text{electric}} = F_{\text{magnetic}} \implies E e = B e v$

Cancelling $e$ on both sides gives the velocity selector equation:

$v = \frac{E}{B} \quad \text{--- (2)}$

Since $E = \frac{V_1}{d}$ (where $d$ is the plate separation), the velocity is precisely known.

Final Derivation of $e/m$ Ratio

Substitute the velocity $v$ from equation (2) into equation (1):

$\frac{E}{B} = \frac{B e r}{m} \implies \frac{e}{m} = \frac{E}{B^2 r}$

$\frac{e}{m} = \frac{V_1}{B^2 r d} \quad \text{(Proved)}$

Conclusion:
By substituting the known values of the electric field $E$ (or potential $V_1$ and plate spacing $d$), the magnetic field induction $B$, and the radius of curvature $r$ of the electron trajectory, the fundamental charge-to-mass ratio ($\frac{e}{m}$) of an electron can be computed precisely.

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