Rod: I = mL²/12 Ring: I' = mR²/2 = m(L/2π)²/2

Substitute R = L/2π

Simplify to get I/I' = 2π²/3

Mass is an intrinsic property

Mass doesn't change with location

Mass remains M regardless of gravity

For pure inductor current lags voltage by 90°

Average power = VI cos φ

When φ = 90°, cos φ = 0, so P = 0

Output power = 100W Input power = 220V×0.5A = 110W

Efficiency = (Output/Input)×100

is equal to (100/110)×100 = 90.9% ≈ 90%

At resonance XL = XC

Net reactance is zero

Therefore current and voltage are in phase

Use Ohm's law: R = V/I where V = 3V, I = 5×10⁻⁴ A

Total R = 3/(5×10⁻⁴) = 6000Ω

Required R = 6000 - 50 = 5950Ω

Use R = R₀[1 + α(T-T₀)] 200 = 100[1 + 0.005(T-100)]

2 = 1 + 0.005(T-100) 1 = 0.005(T-100)

T = 100 + 200 = 400°C

For balance: P/Q = R/S₁ where S₁ is effective resistance

With x in parallel with S: S₁ = (S×x)/(S+x) = (3x)/(3+x)

Solving 2/2 = 2/S₁ gives x = 6Ω

In parallel, voltage V is same across all resistors Using I = V/R For 2Ω: I₁ = V/2

For 3Ω: I₂ = V/3 For 4Ω: I₃ = V/4

Ratio I₁:I₂:I₃ = V/2:V/3:V/4 = 6:4:3

For three cells in same direction: 3E For one cell in opposite: -E Net EMF = 3E - E = 2E

Internal resistance adds regardless of cell direction r₁ + r₂ + r₃ + r₄ = 4r

Therefore equivalent EMF = 2E and total internal resistance = 4r

Power P = VI cosφ where φ is phase difference

Phase difference = π/3, so cosφ = 1/2

Average power = (V₀I₀/2)×(1/2) = V₀I₀/4

Apply vector addition since voltages are phase shifted

Calculate source voltage: Vs = √[(VC-VL)² + VR²]

Substitute: Vs = √[(120-40)² + 60²] = √(6400+3600) = 100V

Calculate impedance of capacitor: XC = 1/ωC

Find RMS voltage: Vrms = 50V

Calculate current: Irms = Vrms/XC = 50mA

Consider complete AC cycle

Note current is sinusoidal and symmetric

Average of complete symmetric cycle is zero

Use Ohm's law: V = IR where I = 5mA = 0.005A

Calculate required total resistance: R = V/I = 20/0.005 = 4000Ω

Required series resistance = 4000 - 50 = 3950Ω

Calculate potential difference across parallel combination

Find currents through each resistor: I₁ = V/R₁, I₂ = V/R₂

Power ratio = V²/R₁ : V²/R₂ = R₂/R₁ = 2:1

Convert time to seconds: 1h 30min = 5400s

Apply Q = It: Q = 20 × 5400

Calculate: Q = 108000 = 10.8×10⁴ C

Define mobility μ = qτ/m where τ is relaxation time

Observe direct relationship between μ and τ

Mobility increases linearly with relaxation time

Calculate phase angle: tanφ = X_L/R = 1/√3

Find φ = 30°

Time lag = φ/2πf = 30°/(2π×50) = 1/600 s

Calculate RMS current: I_rms = P/V = 12/48 = 0.25 A

Peak current = I_rms × √2

I_peak = 0.25 × √2 = 1/(2√2) A

Current for 30 divisions: I = 3/(3000) A

For 20 divisions: I' = (20/30)×I

Calculate required total resistance: R = 3/I' - 50 = 4440Ω

Resistance increases with temperature for metals

Balance condition: R₁/R₂ = l₁/l₂

As R₁ increases, l₁ must increase

Resistance formula: R = ρL/A, where A = πd²/4

Resistance ∝ L and ∝ 1/d²

Maximum R needs maximum L and minimum d

Calculate area: A = πr² = π(0.05×10⁻³)² m²

Convert current to amperes: I = 90×10⁻³ A

Calculate J = I/A = 1.2×10⁷ A/m²

Use transformer ratio equation: N_p/N_s = I_s/I_p

Given N_p:N_s = 1:25 and I_s = 2A

Calculate I_p = 2 × 25 = 50A

Understand difference between peak and RMS values

Note that quoted AC values are typically RMS

Remember V_RMS = V_peak/√2

Consider energy storage in L and C

Note that only R dissipates energy

L and C store energy but don't dissipate

Write Kirchhoff's voltage law equation for the circuit

Apply condition that voltage across r₁ is zero

Solve for R using these conditions

Use power formula: P = VI for AC circuit

Calculate current: I = P/V = 100/220

Simplify to get 5/11 A

Recall color code rules for resistors

Note that absence of tolerance band has specific meaning

Absence of tolerance band indicates 20% tolerance

Understand that Ohm's law states V = IR with R constant

Check which device maintains constant resistance independent of voltage

Among options, only conductors maintain linear V-I relationship

Balance length ∝ terminal potential

V₁/V₂ = l₁/l₂ = 2

Calculate r = 2Ω from ratio

At junction, total incoming current equals outgoing

This represents charge conservation

No charge accumulation at junction

Total impedance Z = √(R² + ω²L²)

Current I = V/Z

Power P = I²R = (V²/Z²)R = V²R/(R² + ω²L²)

R = ρl/A, where A = m/ρl

R ∝ l²/m (substituting A)

Calculate ratios: (5²/1):(3²/3):(1²/5) = 125:15:1

For AC, i = I₀sin(ωt), Irms = I₀/√2

Time to reach peak from rms means sin(ωt) changes from 1/√2 to 1

t = [sin⁻¹(1) - sin⁻¹(1/√2)]/ω = 2.5ms

Current I = neAvd where n is number density

n ≈ 10²⁸ electrons/m³ compensates for small vd

Large n makes current significant

First convert 0.28k to 280Ω. For 4-band code: 1st band gives first digit, 2nd band gives second digit, 3rd band is multiplier, 4th band is tolerance.

Red = 2 (1st digit), Grey = 8 (2nd digit), Brown = ×10¹ (multiplier), Silver = ±10% (tolerance).

These bands in sequence give 280Ω ±10%, which equals 0.28kΩ ±10%

Power factor = cos θ = R/Z = 1/3, where Z is impedance. Also know that tan θ = Xₗ/R, where Xₗ is inductive reactance = 2Ω.

From power factor = 1/3, find θ. Then use tan θ = 2/R to find R.

Since cos θ = 1/3, tan θ = 2/R = √8. Therefore R = √2 Ω

In a potentiometer, voltage varies linearly with length. First calculate potential gradient: 10V/5m = 2V/m. This tells us voltage drop per meter of wire.

At balance point, potential drop across wire equals emf of secondary cell. Balance length is 200cm = 2m.

Calculate emf of secondary cell: (2V/m)(2m) = 4V. This is the potential difference needed to balance secondary cell emf.

Start with the basic equation for terminal voltage: V = E - Ir, where E is emf, I is current drawn, and r is internal resistance. The voltage drops due to internal resistance when current flows.

Carefully substitute the given values: V = 12V - (80A)(2×10⁻² Ω). The second term represents voltage drop across internal resistance.

Calculate: V = 12 - 1.6 = 10.4V. This shows how terminal voltage is less than emf due to internal voltage drop.

When charge moves through potential difference: It gains KE = qV = IVt

It loses PE by same amount = IVt

Without collisions, KE gain equals PE loss. With collisions, energy becomes thermal

Understand that resistance R = ρl/A, where l is length of current path and A is cross-sectional area perpendicular to current. When battery is connected to 1cm × 1/2cm face, current travels 10cm length through smallest area.

Compare all possible configurations: For 1cm × 1/2cm faces: l = 10cm, A = 0.5cm². For 10cm × 1/2cm faces: l = 1cm, A = 5cm². For 10cm × 1cm faces: l = 1/2cm, A = 10cm².

Calculate R ∝ l/A for each case. The first case gives largest l/A ratio, hence maximum resistance.

First find drift velocity: I = neAvd, where n is electron density

vd = I/(neA) = 1/(8×10²⁸×1.6×10⁻¹⁹×5×10⁻⁷)

Time = Length/velocity = 1/vd = 6.4×10³s

Use R = ρl/A. When stretched, l → 2l

By volume conservation, A₁l₁ = A₂l₂, so A → A/2

Therefore R → 4R = 12Ω

Impedance Z = √(R² + (XL - XC)²). Calculate XL = ωL and XC = 1/ωC separately.

XL = 1000×0.9 = 900Ω, XC = 1/(1000×2×10⁻⁶) = 500Ω

Z = √(300² + (900-500)²) = √(90000 + 160000) = 500Ω

Resonance frequency formula is f = 1/(2π√LC). Let's substitute our values carefully.

f = 1/(2π√(10⁻⁴×10⁻⁶)) = 1/(2π√10⁻¹⁰) = 10⁵/2π Hz

Always check units: L in henry, C in farad gives frequency in hertz!

Initially, all energy is electric (½q²/C). As oscillation starts, energy converts between electric and magnetic forms.

Equal energy means ½q²/C = ½LI². This occurs at ¼ period of oscillation.

Period T = 2π√LC, so time for equal energy = T/4 = π√LC/4

RMS value is root mean square over one cycle. Need to integrate (i₁sinωt + i₂cosωt)² over period.

Sin and cos terms are orthogonal - their product averages to zero over complete cycle.

Therefore, I_rms = √[(i₁²/2 + i₂²/2)] = √(i₁² + i₂²)/2

When wire is stretched, length increases by 10% (l₂ = 1.1l₁), and area decreases (A₂ = A₁/1.1) due to constant volume.

Resistance R = ρl/A. New resistance = ρ(1.1l)/((A/1.1)) = 1.21R₁

Specific resistance (ρ) is material property, doesn't change with shape!

Power is proportional to voltage squared (P = V²/R). When voltage changes, power changes quadratically.

If V₂ = 0.975V₁ (2.5% decrease), then P₂/P₁ = (V₂/V₁)² = (0.975)²

Therefore, P₂/P₁ = 0.95, meaning power decreases by 5%. Notice it's double the voltage percentage!

Current and deflection are proportional in galvanometer. If I₁/I₂ = θ₁/θ₂, then (30/20) = (R₂/R₁), where R is total resistance.

Original total R₁ = 3000Ω (2950 + 50). For 20 divisions, new total R₂ = R₁×(30/20) = 4500Ω

Therefore, new series resistance = 4500 - 50 = 4450Ω (subtract galvanometer resistance)

In a series combination, total EMF is sum of individual EMFs. With 10 identical cells, total EMF is 10E.

When we measure across 3 cells, we're only measuring the EMF of those 3 cells.

Therefore, voltmeter reading = 3E. Internal resistance doesn't affect this reading since voltmeter has very high resistance!

When we heat a metal, its resistance increases. In meter bridge, we're comparing unknown resistance with known resistance.

The balancing equation is R/S = L/(100-L), where L is length. If R increases, L must increase to maintain balance.

Therefore, balance point shifts right. Think about it: we need more wire length on left to balance increased resistance!

Current is charge flowing per second. Here, one electron completes 9.4×10¹⁸ revolutions/s. Each revolution carries charge e=1.6×10⁻¹⁹C.

Current = charge/time = (charge per revolution × revolutions per second)

I = (1.6×10⁻¹⁹)(9.4×10¹⁸) = 1.5A. The units work: C/s = A

Think about what properties we need in a resistor: stable resistance, low temperature coefficient, and durability. Alloys like Manganin and Constantan were specifically developed for this!

Pure metals like Cu, Al, or Ag have low resistance and high temperature coefficients - not good for resistors. Semiconductor materials like Ge aren't used for wire-wound resistors.

Manganin (Cu-Mn-Ni alloy), Constantan (Cu-Ni alloy), and Nichrome are perfect because they have high resistivity, low temperature coefficient, and don't oxidize easily.

Drift velocity is related to mobility and electric field strength. The formula is vd = μE, where μ is mobility. Let's identify our values: μ = 2.5×10⁻⁶ m²/V/s, E = 3×10¹⁰ V/m

Simply multiply these numbers: vd = (2.5×10⁻⁶)(3×10¹⁰). Keep track of units: (m²/V/s)(V/m) = m/s

This gives us 7.5×10⁴ m/s. This is reasonable for drift velocity in strong fields!

At resonance: XL=XC

Net reactive impedance = 0

Phase angle = tan⁻¹(0) = 0°

Calculate current per division: Total current range = 5mA = 0.005A, Number of divisions = 50

Find division per ampere: 50 div/0.005A = 10,000 div/A = 1×10⁵ div/A

Verify units match option (div/A)

Current I ∝ drift velocity vd

Mobility μ depends only on temperature

At constant T, μ stays same but vd increases

Work = Power × Time

Power in resistor = I²R

W = I²R × t

Calculate Q-factor using Q=(1/R)√(L/C)

Compare Q-factors for all combinations

Highest Q-factor gives best tuning

Start with basic Ohm's law V=IR. Convert to point form by considering voltage per unit length and current per unit area

Express E=V/l and j=I/A where E is electric field and j is current density

Substitute to get E=ρj, where ρ is resistivity

8 cells forward (16V) + 2 cells reversed (-4V) gives net EMF = 12V

Total resistance = 10 cells × 1Ω + external 10Ω = 20Ω

I = 12V/20Ω = 0.6A

Calculate bulb resistance: R=V²/P=(120)²/60=240Ω

Find current through bulb: I=P/V=60/120=0.5A

Required series resistance=(220-120)/0.5=200Ω

4.7kΩ = 47 × 102Ω

Third band represents multiplier (×102)

Red band indicates multiplier of 100

Grey(8), Red(2) give 82

First gold = ×0.1 multiplier

Second gold = ±5% tolerance