Electric Current Notes

Electric Current

  • If two conductors have a potential difference (e.g., 60V and 30V), connecting them will cause charge to flow from the higher potential to the lower potential.

  • Charge continues to flow until the potential is the same in both conductors.

  • Maintaining a constant potential difference enables continuous charge flow, which constitutes electric current.

  • Electric current is the flow of charge.

Definition of Electric Current

  • Electric current is defined as the rate of flow of charge.

  • It's the rate of flow of charge (dq/dt) through any cross-section of a conductor.

  • This rate is also referred to as instantaneous current.

Instantaneous Current

  • Instantaneous current is mathematically represented as dq/dt.

  • If charge is a function of time, differentiating it yields the current as a function of time.

  • If you differentiate charge with respect to time, you get current at that instant.

  • Unit: Ampere (A), a fundamental quantity with dimension A.

Relationship Between Current and Charge

  • dq = idt

  • Integrating current with respect to time gives the total charge flown through a cross-section.

  • Integrating the current over time gives the charge flow-through that cross-section.

Graphical Interpretation

  • On a QT graph (charge vs. time), the slope of the tangent at any point gives the instantaneous current at that time.

  • In an IT graph (current vs. time), the area under the graph represents the charge flow.

Formulas

  • Instantaneous Current: i = \frac{dq}{dt}

  • Charge Flow: q = \int idt

Example Problems

Problem 1: Finding Current from Charge Function

  • Given a charge-time relation, find the current.

  • Differentiate the charge function with respect to time to find the current function.

  • Substitute the given time to find the current at that instant.

  • Example: If q = 3t^2 + 4t + 5, then i = \frac{dq}{dt} = 6t + 4. At t = 3 seconds, i = 6(3) + 4 = 22 Amperes.

Problem 2: Finding Charge from Current Function

  • Given a current-time relation, find the charge flow.

  • Integrate the current function with respect to time over the given interval.

  • Example: If i = 4t + 3, then q = \int{0}^{5} (4t + 3) dt = [2t^2 + 3t]{0}^{5} = (2(5)^2 + 3(5)) - (0) = 50 + 15 = 65 Coulombs.

Problem 3: Finding Charge from IT Graph

  • If you have a current-time graph, the area gives the charge.

  • Example: From 0 to 5 seconds, the current decreases. Calculate the area of the triangle to find the charge flow.

  • If the current decreases from 10A to 0A over 5 seconds, the charge flow is \frac{1}{2} \cdot 5 \cdot 10 = 25 Coulombs.

Direction of Current

  • Conventional current: The direction of flow of positive charge (historical convention).

  • Electronic current: The direction of flow of electrons (opposite to conventional current).

  • Unless specified, assume conventional current.

  • Electrons flow from the negative terminal to the positive terminal.

Scalar vs. Vector Quantity

  • Current is a scalar quantity, although it has direction.

  • It does not follow vector laws like the parallelogram law or triangle law.

  • Technically, current is a tensor quantity, but it is treated as a scalar in basic contexts.

Charge Carriers

  • In solid conductors (metals), free electrons carry charge.

  • Metals have a large number of free electrons (e.g., 10^{29} per cubic meter).

  • In liquids, ions carry charge (e.g., in solutions like sodium chloride or copper sulfate).

  • Pure water (H2O) does not conduct electricity because it lacks ions.

  • In gases, charge is carried by ions, but only under high electric fields that cause ionization.

Average Current

  • Average current is the total amount of charge passing through any given cross-section in a given time interval.

Charge Flow Problems

  1. Both charges are going the same direction; that can be positive or negative. Therefore, they get added together. If the direction of the charge is in opposite directions from each other, then you need to subtract. When both charges are positive charges, then it results in a plus, whereas, if they both go in the other direction, they will need to be subtracted.

Example Problems

Problem 1: Electron Gun

  • An electron gun emits 2 \times 10^6 electrons per second. What current does this constitute?

  • Charge per second (current) = number of electrons per second * charge of one electron.

  • I = (2 \times 10^6) \times (1.6 \times 10^{-19}) = 3.2 \times 10^{-13} Amperes.

Problem 2: Electron Orbiting

  • An electron is rotating in a circular orbit of radius r with speed V. What is the current?

  • Time period of one revolution: T = \frac{2\pi r}{V}

  • Charge flow through any cross-section in time T is e.

  • Current I = \frac{e}{T} = \frac{e}{\frac{2\pi r}{V}} = \frac{eV}{2\pi r}

Problem 3 (Assignment Problem Hint)

  • If angular velocity \omega is given instead of speed V: Use T = \frac{2\pi}{\omega} for the time period and substitute in the current formula.

Problem 4 (Assignment Problem Hint)

  • Given that 2.5 \times 10^{18} electrons move from A to B and 1.0 \times 10^{18} positive ions move from B to A. To find the Net charge, remember that because they are moving in opposite direction, you must ADD their magnitudes