Specific charge of particles, nuclei and ions

Specific charge is the charge of a particle divided by its mass. Its unit is coulombs per kilogram (C/kg, written C kg⁻¹). The electron has the largest specific charge of any particle you meet at A-level, 1.76 × 10¹¹ C kg⁻¹, because it carries the same size of charge as a proton on about 1/1,840 of the mass.

The AQA A-level Physics specification asks for the specific charge of the proton and the electron, and of nuclei and ions, in its section on the constituents of the atom. Specific charge matters because it decides how strongly a particle is deflected by electric and magnetic fields: a higher specific charge means a bigger acceleration from the same field.

Charges and masses you need

These values are from the AQA data booklet, which you get in the exam.

The neutron has no charge, so its specific charge is zero. Specific charge is usually quoted as a size, but if a question cares about sign, an electron or a negative ion has a negative charge.

Specific charge of an electron

Divide the numbers, 1.60 ÷ 9.11 = 0.1756, and subtract the powers, −19 − (−31) = 12. So 0.1756 × 10¹² = 1.76 × 10¹¹ C kg⁻¹, which matches the booklet.

Specific charge of a proton

The booklet gives 9.58 × 10⁷ C kg⁻¹ because it uses more precise values of the charge and mass. Either is fine in an exam if you show your working. The electron's specific charge is about 1,840 times bigger than the proton's, which is just the ratio of their masses.

Check you can tell the particles apart by charge and mass, then press Mark to check your answer in the course:

Specific charge of a nucleus

A nucleus contains only protons and neutrons. Its charge comes from the protons, and its mass from all the nucleons.

• Charge: number of protons × 1.60 × 10⁻¹⁹ C.
• Mass: number of nucleons × 1.67 × 10⁻²⁷ kg. A-level questions accept this approximation, treating protons and neutrons as the same mass.

Example: a carbon-12 nucleus

Carbon-12 has 6 protons and 12 nucleons.

A helium-4 nucleus, an alpha particle, gives the same answer, because it also has half as many protons as nucleons: 2 out of 4.

Specific charge of an ion

An ion is an atom that has gained or lost electrons. Its charge is the number of electrons lost (positive) or gained (negative) times 1.60 × 10⁻¹⁹ C. For the mass, the electrons are so light that the nucleons are all that matter.

Example: a magnesium-24 ion, Mg²⁺

The ion has lost 2 electrons, so Q = 2 × 1.60 × 10⁻¹⁹ = 3.20 × 10⁻¹⁹ C. It has 24 nucleons, so m = 24 × 1.67 × 10⁻²⁷ = 4.008 × 10⁻²⁶ kg.

Including the ion's 10 remaining electrons adds 10 × 9.11 × 10⁻³¹ = 9.11 × 10⁻³⁰ kg, about 0.02% of the mass, and the answer still rounds to 7.98 × 10⁶ C kg⁻¹.

Example: an oxygen-16 ion, O²⁻

It has gained 2 electrons, so its charge is −3.20 × 10⁻¹⁹ C, and its mass is 16 × 1.67 × 10⁻²⁷ = 2.672 × 10⁻²⁶ kg. The specific charge is 1.20 × 10⁷ C kg⁻¹ in size, negative in sign.

Where specific charge comes from

Cathode rays were the first particles to have their specific charge found: J. J. Thomson bent them with electric and magnetic fields in 1897 and worked out Q/m from how far they moved. The ratio was far larger than for any ion, which showed the particles, electrons, were much lighter than atoms. Robert Millikan's oil drop experiment later measured the charge e itself, and with both, the electron's mass could be found. The Particles course carries on from here to quarks, leptons and the Standard Model.

Common mistakes

• Dividing mass by charge. Specific charge is charge ÷ mass.
• Using relative units. The formula needs coulombs and kilograms, not charge +1 and mass 1 u.
• Counting electrons in a nucleus. A nucleus is protons and neutrons only.
• Using the proton number for an ion's charge. An ion's charge is the electrons gained or lost, not the number of protons.
• Wrong powers of ten. Subtract the indices when dividing: 10⁻¹⁹ ÷ 10⁻³¹ = 10¹².

Read next

Physics · Electricity and magnetismCharge equation Q = It: charge, current and timeThe charge equation Q = It, what charge and current are, units, rearranging, charge and energy with E = QV, electrons per coulomb, examples and practice.