TLDR;
This video provides a comprehensive overview of current electricity, covering essential concepts and formulas. It begins with the definition of electric current, drift velocity, and current density, then moves on to Ohm's law, resistance, and their temperature dependence. The video also explains combinations of resistances and cells, Kirchhoff's laws, and the heating effect of electric current. Finally, it discusses the conversion of galvanometers and Wheatstone bridges.
- Electric current and its properties
- Drift velocity and its relation to current
- Ohm's law and resistance
- Temperature dependence of resistance
- Kirchhoff's laws and circuit analysis
- Heating effect of electric current
- Galvanometer conversion and Wheatstone bridge
Introduction [0:00]
The video introduces the chapter on current electricity, emphasizing that while it is not particularly difficult, it contains many scalar quantities. The content is extensive, similar in size to the electrostatics chapters. The lecture aims to provide a mind map of the key concepts.
Electric current [0:49]
Electric current is defined as the flow of charge. In paper, current is treated as a scalar quantity due to the lack of directional options. The SI unit of current is the ampere (A), equivalent to a coulomb per second. Average current is calculated as the total charge flown divided by the total time taken. Instantaneous current is the slope of the charge versus time (q vs t) graph. The area under the current versus time (I vs t) graph gives the charge flow.
Drift velocity [2:38]
In a conductor without a potential difference, electrons move randomly with high kinetic energy (approximately 10^-21 Joules) due to room temperature. The average speed of these electrons is around 10^5 meters per second, but their average velocity is zero because their motion is random, resulting in no net displacement or current. The mean relaxation path is the average distance an electron travels between collisions (around 10 angstroms), and the mean relaxation time is the average time between collisions (approximately 10^-14 seconds). When a potential difference is applied, an electric field is set up, causing electrons to drift towards the positive terminal with a drift velocity of about 10^-4 meters per second. The formula for drift velocity (vd) is vd = -eEτ/m, where e is the charge of the electron, E is the electric field, τ is the relaxation time, and m is the mass of the electron.
Current & drift velocity relation [6:34]
The relationship between current (I) and drift velocity (vd) is given by the formula I = nAevd, where n is the number of electrons per unit volume, A is the cross-sectional area, and e is the charge of an electron. Mobility (μ) is defined as the drift velocity per unit electric field (μ = vd/E). For a conductor, mobility remains constant if the temperature is not raised.
Current density [8:58]
Current density (J) is defined as the current per unit area and is a vector quantity. Its magnitude is J = I/A, and its direction is the same as the current flow. The unit of current density is amperes per meter squared (A/m²). The relationship between current density and electric field is J = σE, where σ is the conductivity. Conductivity is the inverse of resistivity (ρ), and their product is constant. For a conductor with a uniform cross-section, the electric field and current density are the same everywhere. If the cross-sectional area is not uniform, the current remains the same at all points, but the current density varies inversely with the area.
Ohm’s law [13:23]
Ohm's Law states that the current through a conductor is directly proportional to the potential difference applied across it (V = IR), where R is the resistance. The SI unit of resistance is the ohm (Ω). The slope of the voltage versus current (V vs I) graph gives the resistance. For conductors, resistance increases with temperature. Substances that follow Ohm's Law are called ohmic, while those that do not are non-ohmic (e.g., diodes, semiconductors). The resistance (R) of a conductor is given by R = ρL/A, where ρ is the resistivity, L is the length, and A is the cross-sectional area. Resistivity depends on the material and temperature but not on the dimensions.
Resistance with dimension [17:23]
The resistance of a conductor depends on the direction of current flow. The length (L) is the length of the conductor in the direction of the current, and the cross-sectional area (A) is perpendicular to the current flow. For a circular wire with points A and B, the effective resistance between A and B is given by R_effective = R * θ1 * θ2 / (θ1 + θ2), where R is the resistance per unit length, and θ1 and θ2 are the smaller and larger angles, respectively. For a frustum, the resistance is given by R = ρL / (πab), where a and b are the radii of the ends. For two bodies with the same length and cross-sectional area connected in series, the equivalent resistivity is ρ_eq = 2ρ1ρ2 / (ρ1 + ρ2). For parallel connections, the equivalent resistivity is ρ_eq = (ρ1 + ρ2) / 2.
Cutting & stretching the wire [23:01]
When cutting a wire, the resistance is directly proportional to the length. When stretching a wire, the resistance is directly proportional to the square of the length and inversely proportional to the area. If the change in length or radius is less than 5%, the percentage change in resistance is approximately twice the percentage change in length.
Temperature dependence of resistance [25:23]
The temperature dependence of resistance is given by the formula Rt = R0(1 + αT), where Rt is the resistance at temperature T, R0 is the resistance at 0 degrees Celsius, and α is the temperature coefficient of resistance. For metals, α is positive, and for semiconductors, α is negative. If two bodies have the same resistance, the equivalent α in series or parallel is α_eq = (α1 + α2) / 2.
Combination of resistance [27:50]
In a series combination of resistances, the current is the same, and the potential is divided. The equivalent resistance is the sum of all resistances (R_eq = R1 + R2 + R3 + ...). In a parallel combination, the voltage is constant, and the current is divided. The reciprocal of the equivalent resistance is the sum of the reciprocals of individual resistances (1/R_eq = 1/R1 + 1/R2 + 1/R3 + ...). For two resistors in parallel, the equivalent resistance is R_eq = (R1 * R2) / (R1 + R2).
Current and Voltage divider rule [30:32]
In a series circuit, the voltage is divided in the ratio of the resistances (V1:V2:V3 = R1:R2:R3). In a parallel circuit, the current is divided in the inverse ratio of the resistances.
Kirchhoff’s law & 3D-Circuit [32:04]
For a cube with 12 edges, each having a resistance of R ohms, the equivalent resistance between the edge is 7R/12, between face diagonal is 3R/4, and between body diagonal is 5R/6. Kirchhoff's Laws include the Junction Law and the Voltage Law. The Junction Law is based on the conservation of charge, stating that the incoming current equals the outgoing current at a junction. The Voltage Law is based on the conservation of energy, stating that the sum of the potential differences in any closed loop is zero.
Cell and internal resistance [36:05]
A real cell has internal resistance, which depends on the electrolyte. The terminal potential difference (V) is given by V = E - Ir, where E is the EMF, I is the current, and r is the internal resistance. When current is drawn from the cell, the terminal voltage is less than the EMF. When current is given to the cell, the terminal voltage is greater than the EMF. In an open circuit, the terminal potential difference equals the EMF. The power delivered by a cell during the withdrawal of current is discussed, along with the maximum power transfer theorem, which states that maximum power transfer occurs when the external resistance equals the internal resistance. The maximum power consumed is Esquare/4r.
Combination of cells [41:50]
In a series combination of cells with the same polarity, the equivalent EMF is the sum of the individual EMFs, and the equivalent internal resistance is the sum of the individual internal resistances. The net current is the total EMF divided by the total resistance. In a parallel combination of cells, the equivalent EMF is given by a complex formula involving individual EMFs and internal resistances. If all cells have equal EMFs and internal resistances, the equivalent EMF is the same as the individual EMF. In a mixed combination of cells, the net EMF and current can be calculated based on the number of rows and cells in each row.
Heating effect of electric current [47:30]
The heating effect of electric current is discussed, including problems involving electric kettles. If two resistances take times t1 and t2 to make tea, the time taken in series is t1 + t2, and in parallel is (t1 * t2) / (t1 + t2). For a bulb, the rated power is related to the rated voltage and resistance by P = V^2/R. In a series connection of bulbs, the brightness is inversely proportional to the rated power. In a parallel connection, the brightness is directly proportional to the rated power.
Conversions of Galvanometer [52:42]
A galvanometer can be converted into an ammeter by adding a small shunt resistance in parallel. The shunt resistance (s) is given by s = (Ig * G) / (I - Ig), where Ig is the full-scale deflection current and G is the galvanometer resistance. A galvanometer can be converted into a voltmeter by adding a very high resistance in series. The range of the voltmeter is V = Ig * (G + R).
Wheatstone bridge [56:00]
In a balanced Wheatstone bridge, the condition is R1/R3 = R2/R4. The potential of points P and Q is the same, and the current through the galvanometer is zero.
Meter bridge [56:58]
The Meter Bridge is based on the balanced Wheatstone bridge principle and is used to find unknown resistance. The unknown resistance (x) is given by x = R * (100 - l) / l, where l is the balancing length.
Thankyou bachhon [59:47]
The video concludes with a thank you message, encouraging viewers to keep the discussed points in mind and create short notes for revision. The next mind map will cover the magnetic effect of current.