Saksham Study Point

Electric Current

Basics

I = q / t

Average current: Iav = Δq / Δt
Instantaneous current: Iinst = dq / dt
I-t graph: Iav = (area under graph) / (total time)
Concave area = 2/3 (x0y0), Convex area = 1/3 (x0y0)

Drift Velocity

Electron Flow

Electrons random motion: net average ≈ 0
Electric field gives drift (slow ordered motion)
vd = aτ

vd = (eEτ) / m

Relation: I = n A e vd

Factors Affecting Drift Velocity

Shape + Field

vd ∝ E
Uniform shape: E1 = E2 = E3 ⇒ vd1 = vd2 = vd3
Non-uniform shape: E1 > E2 > E3 ⇒ vd1 > vd2 > vd3
Current vs drift: I = nAVde

Ohm’s Law

V-I

V = I R

V-I graph slope = R (tanθ)
Non-ohmic conductor: graph not linear, R not constant
Resistance depends on material, temperature, dimension

R = ρl / A

Current Density

J

J = I / A

Also: J = nevd
Relation: J = E / ρ and E = ρJ
Uniform cross-section: current same everywhere
Non-uniform cross-section: J varies (area changes)

Dependence of R on Dimension

ρ, l, A

R = ρl/A
Longer wire ⇒ R increases
Thicker wire (more area) ⇒ R decreases

Cutting & Stretching of Wire

Tricks

Stretching (length becomes n times) ⇒ R becomes n²R
If radius becomes r/n ⇒ R becomes n⁴R
If change in length < 10%: % change in R ≈ 2 × % change in length

Temperature Dependence of Resistance

α

Metals: α positive, T↑ ⇒ R↑
Semiconductors: α negative, T↑ ⇒ R↓
Alloys: α small (R changes slightly)

Equivalent α: (α1R1 + α2R2) / (R1 + R2)

Grouping of Resistance

Series / Parallel

Series: current constant, voltage divides
Rs = R1 + R2 + ...
Parallel: voltage constant, current divides
1/Rp = 1/R1 + 1/R2 + ...
Two resistors parallel shortcut: Rp = (R1R2)/(R1+R2)

Current & Voltage Divider Rule

Rules

Current divider (2 parallel): I1 = I·R2/(R1+R2), I2 = I·R1/(R1+R2)
Voltage divider (series): V1 = V·R1/(R1+R2+R3)
Similarly V2, V3… proportional to resistance

Kirchhoff’s Law

Junction + Loop

Junction rule: Iin = Iout
Loop rule: algebraic sum of potential changes = 0
Common terms: +E, -E, +IR, -IR depending on direction

Cell & Internal Resistance

TPD

When current drawn: V = E − Ir
Open circuit: I = 0 ⇒ TPD = E
Short circuit: Imax = E/r, TPD = 0

I = E / (R + r)

Maximum Power Transfer

Condition

Condition: R = r
P = I²R

Pmax = E² / (4r)

Combination of Cells

Series / Parallel / Mixed

Series: Eeq = E1+E2+..., req = r1+r2+...
If all same: i = nE/(nr + R)
Parallel (general): Eeq = (E1/r1 + E2/r2 + ...)/(1/r1+1/r2+...)
Parallel (same cells): Eeq = E, req = r/n, i = E/(R + r/n)
Mixed: m rows, n in series each row (internal per row = nr)

Pmax (same cells, series): nE² / (4r) (when R = nr)

Wheatstone Bridge

Balance

Balanced: current through galvanometer = 0
Condition: R1/R2 = R3/R4
Balanced ⇒ VP = VQ

Meter Bridge

Practical

Used to find unknown resistance using balance point
Based on Wheatstone bridge balance condition

Potentiometer

Gradient

Potential gradient: k = V/L
Compare cells: E1/E2 = l1/l2
Internal resistance: r = R( (l1/l2) − 1 )
Same polarity: E1l1 = E2l2, opposite polarity: E1l1 = -E2l2

Heating Effect & Power

Joule

P = VI = I²R = V²/R
Heat: H = (I²R)t
Brightness proportional to dissipated power

Combination of Bulbs

Series / Parallel

Series: same current, power ∝ R, brightness ∝ R
Parallel: same voltage, higher power bulb glows more
If one bulb fused in series: others off / changes as per circuit

Galvanometer Conversion

Ammeter / Voltmeter

To ammeter: shunt S in parallel
S = (Ig G) / (I − Ig) , where n = I/Ig ⇒ S = G/(n−1)
Resistance of ammeter = G + S (as given)
To voltmeter: series resistance R
R = (V/Ig) − G
Ideal voltmeter resistance = ∞, ideal ammeter resistance = 0
Sensitivity: Si = θ/I, Sv = θ/V