Specific charge is the ratio of charge to mass

Atomic Structure & Nuclear Stability — AQA A-Level Physics

Key Definition

Specific charge — The ratio of the total charge of a particle to its mass. Measured in C kg⁻¹.

\text{specific chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C).} = \frac{Q}{m}

$Q$: total chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). of the particle (C)
$m$: total mass of the particle (kg)

The specific chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). of the electron $= e/m_e = 1.76 \times 10^{11}$ C kg$^{-1}$.
The specific charge of the proton $= e/m_p = 9.58 \times 10^{7}$ C kg$^{-1}$.
Both values are on the data sheet as 'charge/mass ratio'.
The electron's specific charge is about 1800 times larger than the proton's because its mass is much smaller.

Calculating specific charge of ions and nuclei

For an ion: charge $=$ number of electrons gained/lost $\times 1.60 \times 10^{-19}$ C. Mass $=$ nucleon number $\times 1.67 \times 10^{-27}$ kg.
For a nucleus: charge $=$ number of protons $\times 1.60 \times 10^{-19}$ C. Mass $=$ nucleon number $\times 1.67 \times 10^{-27}$ kg.
If an ion gains electrons, the specific charge is negative. If it loses electrons, positive.

Worked Example

An atom of ${}_{12}^{24}\text{Mg}$ gains 2 electrons. What is the specific charge of the ion?

Show Solution

List known values

Charge of an electron $= -1.60 \times 10^{-19}$ C
Mass of a nucleon $= 1.67 \times 10^{-27}$ kg
Nucleon number $A = 24$

Calculate the total mass

$$m = 24 \times 1.67 \times 10^{-27} = 4.0 \times 10^{-26} \text{ kg}$$

Calculate the total charge

The atom was neutral before gaining 2 electrons, so:

$$Q = 2 \times (-1.60 \times 10^{-19}) = -3.2 \times 10^{-19} \text{ C}$$

Calculate specific charge

$$\text{specific charge} = \frac{Q}{m} = \frac{-3.2 \times 10^{-19}}{4.0 \times 10^{-26}} = -8.0 \times 10^{6} \text{ C kg}^{-1}$$

Answer

Specific charge $= -8.0 \times 10^{6}$ C kg$^{-1}$

Common Mistake MEDIUM

Students often: Forgetting the sign on specific charge for ions.
Instead: Gain of electrons gives a negative specific charge. Loss of electrons gives a positive specific charge. Always include the sign.