Current Electricity — JEE Mains Physics

Formulas (2)

• Charge, current and the area under an I–t graph · Charge and current
  $$I = \frac{dq}{dt}, \qquad q = \int_{t_1}^{t_2} I\,dt, \qquad n = \frac{I}{e}$$
• Drift velocity, mobility and current density · Drift and current density
  $$I = neAv_d, \qquad v_d = \frac{eE\tau}{m} = \mu E, \qquad J = nev_d = \sigma E$$

Watch out for (6)

• Use the limits the question gives
  → Charge, current and the area under an I–t graph
• mAh is not coulombs
  → Charge, current and the area under an I–t graph
• Least current comes from dI/dt, not dq/dt
  → Charge, current and the area under an I–t graph
• mm² is 10⁻⁶ m²
  → Drift velocity, mobility and current density
• Electrons drift towards higher potential
  → Drift velocity, mobility and current density
• At fixed voltage the area does not matter
  → Drift velocity, mobility and current density

Formulas (3)

• Resistivity and the shape of a conductor · Resistance from shape
  $$R = \frac{\rho l}{A}, \qquad A_{\text{hollow}} = \pi\left(r_2^{2} - r_1^{2}\right), \qquad \sigma = \frac{1}{\rho}$$
• Stretching, melting and redrawing a wire · Constant-volume stretching
  $$R' = n^{2}R \ (\text{length} \times n), \qquad R \propto \frac{1}{r^{4}}, \qquad \frac{\Delta R}{R} \approx 2\frac{\Delta l}{l}$$
• Resistance and temperature · Temperature dependence
  $$R_T = R_0\left(1 + \alpha\,\Delta T\right), \qquad t = \frac{R_t - R_0}{R_{100} - R_0} \times 100\ ^{\circ}\text{C}$$

Watch out for (9)

• Radius, not diameter
  → Resistivity and the shape of a conductor
• Resistivity does not change with shape
  → Resistivity and the shape of a conductor
• Which faces?
  → Resistivity and the shape of a conductor
• R goes as n², not n
  → Stretching, melting and redrawing a wire
• 2Δl/l is only for small changes
  → Stretching, melting and redrawing a wire
• "By twice" is three times
  → Stretching, melting and redrawing a wire
• Know the reference temperature
  → Resistance and temperature
• Kelvin at the end
  → Resistance and temperature
• A hotter wire draws less current
  → Resistance and temperature

Formulas (2)

• Series, parallel and bent wires · Series, parallel, loops
  $$R_s = \sum R_i, \qquad \frac{1}{R_p} = \sum \frac{1}{R_i}, \qquad R_{\text{loop}} = R\,x(1 - x)$$
• Redrawing networks: shorts and symmetry · The two moves
  $$V_P = V_Q \ \Rightarrow\ I_{PQ} = 0, \qquad R_{\text{eq}} = \frac{V_{\text{in}} - V_{\text{out}}}{I_{\text{total}}}$$

Watch out for (6)

• A cut wire divides twice
  → Series, parallel and bent wires
• Arcs go by length
  → Series, parallel and bent wires
• Check every option numerically
  → Series, parallel and bent wires
• A missed short
  → Redrawing networks: shorts and symmetry
• A crossing is not a junction
  → Redrawing networks: shorts and symmetry
• Symmetry must be about the input–output line
  → Redrawing networks: shorts and symmetry

Formulas (3)

• Current and voltage dividers · Dividers
  $$V_1 = V\frac{R_1}{R_1 + R_2}, \qquad I_1 = I\frac{R_2}{R_1 + R_2}$$
• The balanced Wheatstone bridge · Balance condition
  $$\frac{P}{Q} = \frac{R}{S} \ \Rightarrow\ I_{\text{middle}} = 0, \qquad R_{\text{eq}} = \frac{(P + Q)(R + S)}{P + Q + R + S}$$
• Kirchhoff's laws and node potentials · Junction equation at a node x
  $$\sum_i \frac{V_x - V_i}{R_i} = 0, \qquad \sum \varepsilon = \sum IR$$

Watch out for (9)

• Current divides inversely
  → Current and voltage dividers
• A rating is not the working voltage
  → Current and voltage dividers
• A load lowers the output
  → Current and voltage dividers
• Pair the arms correctly
  → The balanced Wheatstone bridge
• The middle arm's value is irrelevant at balance
  → The balanced Wheatstone bridge
• Check before assuming balance
  → The balanced Wheatstone bridge
• The sign of a cell depends on the direction you cross it
  → Kirchhoff's laws and node potentials
• A negative current is not a mistake
  → Kirchhoff's laws and node potentials
• One equation per unknown node
  → Kirchhoff's laws and node potentials

Formulas (2)

• EMF, internal resistance and terminal voltage · A real cell
  $$I = \frac{\varepsilon}{R + r}, \qquad V = \varepsilon \mp Ir, \qquad P_{\max} = \frac{\varepsilon^{2}}{4r}\ (R = r)$$
• Cells in series, parallel and opposition · Two cells in parallel
  $$\varepsilon_{\text{eq}} = \frac{\varepsilon_1/r_1 + \varepsilon_2/r_2}{1/r_1 + 1/r_2}, \qquad r_{\text{eq}} = \frac{r_1r_2}{r_1 + r_2}$$

Watch out for (6)

• Charging adds Ir
  → EMF, internal resistance and terminal voltage
• Maximum power is at R = r, not R = 0
  → EMF, internal resistance and terminal voltage
• Include r in the total
  → EMF, internal resistance and terminal voltage
• Weight the emfs by 1/r
  → Cells in series, parallel and opposition
• A reversed cell still adds its resistance
  → Cells in series, parallel and opposition
• Equivalent emf lies between the emfs
  → Cells in series, parallel and opposition

Formulas (3)

• Real voltmeters and ammeters · Loaded reading
  $$V_{\text{read}} = V\,\frac{R \parallel R_V}{R \parallel R_V + R_{\text{rest}}}, \qquad I_A = I\,\frac{S}{S + R_A}$$
• The meter bridge · Balance
  $$\frac{P}{Q} = \frac{l}{100 - l}$$
• The potentiometer · Balance and internal resistance
  $$\varepsilon = kl, \qquad k = \frac{V_{\text{wire}}}{L}, \qquad r = R\,\frac{l_1 - l_2}{l_2}$$

Watch out for (9)

• A real voltmeter reads low
  → Real voltmeters and ammeters
• Higher meter resistance is better
  → Real voltmeters and ammeters
• The slope is not R
  → Real voltmeters and ammeters
• Measure from P's end
  → The meter bridge
• Which way does a shunt move the null point?
  → The meter bridge
• Wire properties drop out
  → The meter bridge
• The wire gets only its share
  → The potentiometer
• Shunted means terminal voltage
  → The potentiometer
• Longer wire, more sensitive
  → The potentiometer

Formulas (2)

• Power ratings, bulbs and heaters · From a rating
  $$R = \frac{V_0^{2}}{P_0}, \qquad P = P_0\left(\frac{V}{V_0}\right)^{2}, \qquad R_s = \frac{V - V_0}{P_0/V_0}$$
• Joule heating, losses and efficiency · Joule's law
  $$H = I^{2}Rt = VIt = \frac{V^{2}t}{R}, \qquad Q = mc\,\Delta T + mL$$

Watch out for (6)

• The resistance stays, the power changes
  → Power ratings, bulbs and heaters
• In series the lower rating glows more
  → Power ratings, bulbs and heaters
• Power goes as the square
  → Power ratings, bulbs and heaters
• In parallel the smaller resistor heats more
  → Joule heating, losses and efficiency
• "Whole circuit" includes r
  → Joule heating, losses and efficiency
• Warm before you melt
  → Joule heating, losses and efficiency

Formulas (3)

• Capacitors in steady state · Steady state
  $$I_{\text{C branch}} = 0, \qquad Q = CV_C, \qquad U = \tfrac{1}{2}CV_C^{2}$$
• Just after switching and long after · The two moments
  $$t = 0^{+}: \ L \to \text{open},\ C \to \text{wire}; \qquad t \to \infty: \ L \to \text{wire},\ C \to \text{open}$$
• Charging, discharging and the time constant · Exponential change
  $$q = q_0e^{-t/RC}, \qquad i = \frac{E}{R}\left(1 - e^{-tR/L}\right), \qquad t_{1/n} = \tau\ln n$$

Watch out for (9)

• The series resistor drops nothing
  → Capacitors in steady state
• Leave the capacitor branch out of the equivalent resistance
  → Capacitors in steady state
• The capacitor voltage is a difference of potentials
  → Capacitors in steady state
• Which element is the wire, and when
  → Just after switching and long after
• A real inductor keeps its resistance
  → Just after switching and long after
• Nothing jumps
  → Just after switching and long after
• Energy decays twice as fast
  → Charging, discharging and the time constant
• Growth uses 1 − e^(−t/τ)
  → Charging, discharging and the time constant
• ln, not log₁₀
  → Charging, discharging and the time constant