Study sheet: Electric Current and Circuits

📋 Course Outline

1. Electric Charge and Current
2. Charge Flow and Drift Speed
3. Quasi-Static Circuit Approximation
4. Circuit Topology Vocabulary
5. Kirchhoff Node Law
6. Kirchhoff Mesh Law

📖 1. Electric Charge and Current

🔑 Key Concepts & Definitions

• Electric charge : the electrical quantity carried by matter particles, with the quantized value $q = Ze$
• Electric current : the rate of change of charge through a section, given by $I = \frac{dq}{dt}$

★ Must-know

📌 The conventional current direction is the direction of positive-charge motion and is opposite to the motion of negative charges.

Further detail

• A proton has charge $+e$, a sodium ion Na⁺ has charge $+e$, a silicon ion Si²⁻ has charge $-2e$, and a semiconductor hole has charge $+e$.

💡 Memory Hook

Positive charge and negative charge move in opposite conventional-current directions.

📖 2. Charge Flow and Drift Speed

★ Must-know

📐 Formula — For n charge carriers of elementary charge e crossing a conductor section S at speed v, the current magnitude is $I = envS$.

📐 Formula — The carrier drift speed is $v = \frac{I}{enS}$.

• The drift speed is approximately $10^{-5}\ \mathrm{m\,s^{-1}}$, the thermal speed is approximately $10^6\ \mathrm{m\,s^{-1}}$, and the propagation speed is approximately $c = 3\times10^8\ \mathrm{m\,s^{-1}}$.

Further detail

📐 Formula — For a circular conductor of diameter D, the cross-sectional area is $S = \pi\left(\frac{D}{2}\right)^2$.

💡 Memory Hook

Charge flow through a cross-section causes current, while the resulting drift remains slow.

📖 3. Quasi-Static Circuit Approximation

🔑 Key Concepts & Definitions

• ARQS : The quasi-static approximation assumes that the information propagation time τ is negligible compared with the characteristic variation time T, so electrical changes are treated as instantaneous throughout the circuit.

★ Must-know

📐 Formula — For a circuit of length L and signal speed c, the propagation time is $\tau = \frac{L}{c}$ and the ARQS condition is $\tau \ll T$.

📐 Formula — When the characteristic variation period is $T = \frac{1}{f}$, the ARQS condition becomes $L \ll \frac{c}{f}$.

Further detail

• At 50 Hz, $c/f = 6\times10^6\ \mathrm{m} = 6000\ \mathrm{km}$, so the ARQS is valid when the circuit length is much smaller than this scale and is generally used below about 600 km.

💡 Memory Hook

Propagation time τ ≪ variation time T → instantaneous circuit response.

📖 4. Circuit Topology Vocabulary

🔑 Key Concepts & Definitions

• Node : a point of a circuit where connected wires meet
• Mesh : a closed part of a circuit that does not pass through the same node more than once
• Branch : a portion of a circuit between two nodes

📝 Essential Points

📌 Two dipoles are in series when they carry the same current, and they are in parallel when they are subject to the same voltage.

📖 5. Kirchhoff Node Law

📝 Essential Points

📌 In the quasi-static approximation, the sum of currents entering a node equals the sum of currents leaving it: $\sum i_{\mathrm{in}} = \sum i_{\mathrm{out}}$.

• The node law expresses conservation of electric charge.

💡 Memory Hook

Charge conservation at a node → incoming current equals outgoing current.

📖 6. Kirchhoff Mesh Law

★ Must-know

📐 Formula — Voltage additivity gives $U_{AC} = U_{AB} + U_{BC}$.

📌 In the quasi-static approximation, the algebraic sum of the voltages around a mesh in a chosen traversal direction is zero: $\sum U = 0$.

Further detail

• For a mesh with the stated voltage orientation, the relation is $u_3 = u_1 + u_2 + u_4$.

💡 Memory Hook

Traverse a mesh, add signed voltages, obtain zero.

📊 Synthesis Tables

Series and Parallel Dipoles

Arrangement	Shared quantity	Defining condition
Series	Current	The dipoles carry the same current
Parallel	Voltage	The dipoles are subject to the same voltage