Conservation of charge Knowledge-Based Study

Study notes

1. The Fundamental Principle

   The law of conservation of electric charge states that the total electric charge of an isolated system remains constant over time, regardless of the physical or chemical processes taking place within the system. In any closed system, charge can neither be created nor destroyed; it can only be transferred from one body to another or redistributed within a body.

   Electric charge is an intrinsic property of fundamental particles. The net charge of any system is the algebraic sum of all positive and negative charges:

   $$q_{\text{net}} = \sum q_i = q_1 + q_2 + \dots + q_n$$

   If a system undergoes an interaction, the initial net charge must equal the final net charge:

   $$q_{\text{initial}} = q_{\text{final}}$$

   • Positive Charge: Arises from a deficit of electrons relative to protons.
   • Negative Charge: Arises from an excess of electrons relative to protons.
   • Isolated System: A physical region where no matter or charged particles enter or leave.

   How Charge Moves While Conserving Net Charge

   In macroscopic objects, charging processes do not create new electrons or protons. Instead, they involve the movement of mobile charge carriers, predominantly electrons.

   1. Charging by Friction (Triboelectric Effect)

   When two neutral insulators of different materials are rubbed together, thermal and mechanical energy overcome the work function of one material, causing electrons to migrate to the other.

   • The object that gains electrons acquires a net negative charge ($-q$).
   • The object that loses an identical number of electrons acquires an equal magnitude positive charge ($+q$).
   • The total charge remains zero: $q_{\text{net}} = (+q) + (-q) = 0$.

   2. Charging by Conduction (Contact)

   When a charged conductor touches another conductor, free electrons flow between them driven by electrostatic repulsion until electrostatic equilibrium is achieved. The total initial charge of the conductors equals the sum of their final charges.

   3. Charging by Induction

   When a charged object is brought near a neutral conductor without physical contact, charges within the conductor redistribute. If the conductor is temporarily grounded, electrons may enter or leave via the ground wire. When considering the conductor plus the Earth as a combined closed system, total charge is strictly conserved.

   Calculating Charge Sharing Among Conducting Spheres

   When identical conducting spheres are brought into contact and separated, the total charge divides equally among them due to symmetry and mutual electrostatic repulsion.

   For two identical spherical conductors carrying initial charges $q_1$ and $q_2$, the charge on each sphere after contact is:

   $$q' = \frac{q_1 + q_2}{2}$$

   For spheres of different radii $r_1$ and $r_2$, electrical potential equalizes upon contact ($V_1 = V_2$), which yields:

   $$\frac{k q'_1}{r_1} = \frac{k q'_2}{r_2} \implies \frac{q'_1}{r_1} = \frac{q'_2}{r_2}$$Together with the conservation condition $q'_1 + q'_2 = q_1 + q_2$, the charges distribute in direct proportion to their radii:$$q'_1 = \left(\frac{r_1}{r_1 + r_2}\right) q_{\text{total}}, \quad q'_2 = \left(\frac{r_2}{r_1 + r_2}\right) q_{\text{total}}$$

   Example

   Sphere A carries a charge of $+6\text{ }\mu\text{C}$ and identical Sphere B carries $-2\text{ }\mu\text{C}$. When placed in contact and separated:

   $$q_{\text{total}} = (+6\text{ }\mu\text{C}) + (-2\text{ }\mu\text{C}) = +4\text{ }\mu\text{C}$$$$q'_A = q'_B = \frac{+4\text{ }\mu\text{C}}{2} = +2\text{ }\mu\text{C}$$The initial net charge ($+4\text{ }\mu\text{C}$) is exactly equal to the final net charge ($+2\text{ }\mu\text{C} + 2\text{ }\mu\text{C} = +4\text{ }\mu\text{C}$).

   Microscopic and Subatomic Verification

   Charge conservation is not merely an approximation of macroscopic electrostatics; it is a fundamental symmetry of nature obeyed in all known quantum and nuclear reactions.

   1. Pair Production and Annihilation

   High-energy photons can transform into matter and antimatter pairs:

   $$\gamma \longrightarrow e^- + e^+$$

   • Initial charge of photon: $q = 0$
   • Final charge of electron ($-e$) and positron ($+e$): $q = (-e) + (+e) = 0$

   Conversely, in electron-positron annihilation, matter converts into photons, and the net charge remains zero throughout.

   2. Radioactive Beta Decay

   During negative beta decay ($\beta^-$), a neutron decays into a proton, an electron, and an electron antineutrino:

   n \longrightarrow p^+ + e^- + ar{\nu}_e

   • Initial charge (neutron): $0$
   • Final charge: $(+1) + (-1) + 0 = 0$

   In every interaction, whether classical or quantum, electric charge is rigorously conserved.