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SJ PATTERN
September 18, 2024
Current Electricity Formulas
Basic Formulas
Ohm's Law:
V = I * R
Where V is the voltage (in volts), I is the current (in amperes), and R is the resistance (in ohms).
Resistance in Series:
R_total = R_1 + R_2 + R_3 + ...
Resistance in Parallel:
1 / R_total = 1 / R_1 + 1 / R_2 + 1 / R_3 + ...
Electric Power:
P = V * I
Where P is the power (in watts).
Power in Terms of Resistance:
P = I^2 * R
P = V^2 / R
Joule’s Law of Heating:
H = I^2 * R * t
Where H is the heat produced (in joules), t is the time (in seconds).
Current:
I = Q / t
Where Q is the charge (in coulombs) and t is the time (in seconds).
Electromotive Force (EMF) and Potential Difference:
EMF = V + I * R_internal
Where R_internal is the internal resistance of the source.
Additional Formulas
Resistivity:
R = ρ * (L / A)
Where ρ is the resistivity of the material (in ohm-meters), L is the length of the conductor (in meters), and A is the cross-sectional area (in square meters).
Resistivity and Temperature:
ρ_T = ρ_0 * [1 + α * (T - T_0)]
Where ρ_T is the resistivity at temperature T, ρ_0 is the resistivity at reference temperature T_0, and α is the temperature coefficient of resistance.
Kirchhoff's Current Law (KCL):
Sum of currents entering a junction = Sum of currents leaving the junction.
Kirchhoff's Voltage Law (KVL):
Sum of all voltages around a closed loop = 0.
Combination of Cells:
Series Combination:
EMF_total = EMF_1 + EMF_2 + ...
Parallel Combination:
1 / R_total = 1 / R_1 + 1 / R_2 + ...
Internal Resistance of a Cell:
V = EMF - I * r_internal
Drift Velocity:
v_d = I / (n * A * e)
Where v_d is the drift velocity, n is the number density of charge carriers, A is the cross-sectional area, and e is the charge of an electron.
Current Density:
J = I / A
Where J is the current density (in A/m²).
Advanced Formulas
Capacitance of a Parallel Plate Capacitor:
C = ε_0 * (A / d)
Where C is the capacitance (in farads), ε_0 is the permittivity of free space (8.854 × 10^-12 F/m), A is the area of the plates (in square meters), and d is the separation between the plates (in meters).
Capacitance of a Spherical Capacitor:
C = 4 * π * ε_0 * (r_1 * r_2) / (r_2 - r_1)
Where r_1 and r_2 are the radii of the inner and outer spheres, respectively.
Energy Stored in a Capacitor:
E = 1 / 2 * C * V^2
Where E is the energy (in joules), C is the capacitance (in farads), and V is the voltage across the capacitor (in volts).
Charging and Discharging of a Capacitor (RC Circuit):
Charging:
V(t) = V_0 * (1 - e^(-t / (RC)))
Discharging:
V(t) = V_0 * e^(-t / (RC))
Where V_0 is the initial voltage, R is the resistance (in ohms), C is the capacitance (in farads), and t is the time (in seconds).
Power in a Capacitor (AC Circuit):
P = V_rms * I_rms * cos(φ)
Where φ is the phase difference between the voltage and the current.
For an Alternating Current (AC) Circuit:
RMS Voltage:
V_rms = V_0 / √2
RMS Current:
I_rms = I_0 / √2
Where V_0 and I_0 are the peak voltages and currents.
Impedance of an AC Circuit:
Z = √(R^2 + (X_L - X_C)^2)
Where X_L is the inductive reactance (X_L = ωL), and X_C is the capacitive reactance (X_C = 1 / (ωC)).
Phase Angle in AC Circuits:
tan(φ) = (X_L - X_C) / R
Magnetic and Inductive Concepts
Magnetic Field due to Current:
Ampère's Circuital Law:
∮ B · dl = μ_0 I_enc
Magnetic Field Inside a Long Straight Wire:
B = μ_0 * I / (2 * π * r)
Magnetic Field Inside a Solenoid:
B = μ_0 * n * I
Where n is the number of turns per unit length.
Self-Inductance:
Self-Inductance of a Solenoid:
L = μ_0 * (N^2 * A) / l
Where N is the number of turns, A is the cross-sectional area, and l is the length of the solenoid.
Mutual Inductance:
M = Φ_21 / I_1
Where Φ_21 is the magnetic flux through coil 2 due to current I_1 in coil 1.
Inductive Reactance (AC Circuits):
X_L = ω * L
Where ω is the angular frequency (ω = 2 * π * f).
Capacitive Reactance (AC Circuits):
X_C = 1 / (ω * C)
Energy Stored in an Inductor:
E = 1 / 2 * L * I^2
Displacement Current:
I_d = ε_0 * dΦ_E / dt
Where Φ_E is the electric flux.
Faraday's Law of Induction:
∮ E · dl = - dΦ_B / dt
Where Φ_B is the magnetic flux.
Lenz's Law:
The direction of induced current opposes the change in magnetic flux that produced it.
Biot-Savart Law:
dB = (μ_0 / 4π) * (I dl × r̂) / r^2
Where dB is the infinitesimal magnetic field due to a current element I dl at a distance r.
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