Questions: Current Electricity

Q1. Define current density. Show that current density is a vector while current is a scalar.

Answer (English): Current density at a point is the current per unit area flowing through a surface perpendicular to the direction of flow. Its direction is that of the conventional current, so it is a vector. Current is the total charge crossing a surface per unit time, a scalar formed by integrating the current density over the area. In a wire the current density is uniform over the cross section, and the current equals the product of current density and area.

Hinglish: Current density ek point par current per unit area hai, aur iski direction conventional current ki direction hoti hai, isliye yeh vector hai. Current total charge per unit time hai, jo area par integrate karke milta hai, aur scalar hota hai. Wire mein current density uniform hoti hai, aur current J × A hota hai.

Q2. Derive the relation between drift velocity and current.

Answer (English): Let a conductor have n free electrons per unit volume, each of charge e, and let the drift speed be v. In a time dt every electron moves a distance v dt, so the charge crossing a section of area A is the charge contained in a cylinder of length v dt, which is n e A v dt. Divide by dt to get the current, I = n e A v. Current density follows by dividing by A, giving J = n e v.

Hinglish: Conductor mein n free electrons per unit volume hain, har ek ka charge e hai, aur drift speed v hai. Time dt mein electron v dt distance chalta hai, isliye area A wale section se guzarne wala charge cylinder ke andar ka charge hota hai, n e A v dt. Isko dt se divide karo to I = n e A v milta hai, aur area se divide karke J = n e v milta hai.

Q3. State Ohm's law. What are its limitations?

Answer (English): Ohm's law says that at constant temperature the current through a conductor is directly proportional to the potential difference across it, so V = I R. Its limitations matter in exams: it holds only for ohmic materials, only over a limited range of current, and only when the temperature stays fixed. A diode conducts in one direction only, a filament lamp changes resistance as it heats, and an electrolyte does not show a straight line characteristic. So Ohm's law is an empirical rule for certain conductors, not a universal law.

Hinglish: Ohm's law kehta hai ki constant temperature par conductor ke through current uske voltage ke proportional hota hai, V = I R. Iski limits hain: yeh sirf ohmic materials ke liye, current ki limited range ke liye aur constant temperature par chalta hai. Diode ek hi direction mein current leta hai, filament lamp garam hokar resistance badalta hai, aur electrolyte ka graph straight line nahi hota. Isliye Ohm's law universal nahi, sirf kuch conductors ke liye empirical rule hai.

Q4. Why does the resistivity of a metal increase with temperature while that of a semiconductor decreases?

Answer (English): In a metal the number of free charge carriers is fixed, so the current is limited by how often the electrons collide with the vibrating lattice. At higher temperature the lattice vibrates more strongly, collisions become more frequent, the relaxation time falls and the resistivity rises. In a semiconductor the number of carriers is small at ordinary temperature and grows quickly as thermal energy frees more electrons. That increase in carriers outweighs the extra collisions, so the resistivity falls as the temperature rises.

Hinglish: Metal mein free electrons ki sankhya lagbhag fixed hoti hai, isliye current collisions par depend karta hai. Temperature badhne par lattice zyada vibrate karta hai, collisions badh jate hain, relaxation time ghat jata hai aur resistivity badh jati hai. Semiconductor mein charge carriers kam hote hain aur temperature badhne par tezi se badh jate hain, aur yeh badhotri collisions ke effect se zyada hoti hai, isliye resistivity ghatti hai.

Q5. Derive the expression for the resistance of a conductor in terms of its resistivity.

Answer (English): Let the conductor have length l, area A and carry current I under a potential difference V. The field inside is E = V / l and the current density is J = I / A. In the free electron model J = sigma E, so I / A = sigma V / l. Rearrange to get V = I l / (sigma A). Comparing with V = I R gives R = l / (sigma A), and since sigma = 1 / rho, R = rho l / A.

Hinglish: Conductor ki length l, area A, current I aur voltage V lo. Andar field E = V / l aur current density J = I / A. Free electron model se J = sigma E, isliye I / A = sigma V / l. Rearrange karke V = I l / (sigma A) milta hai. V = I R se compare karo to R = l / (sigma A), aur sigma = 1 / rho rakhne par R = rho l / A mil jata hai.

Q6. A cell has emf E and internal resistance r. Derive the expression for its terminal voltage when it drives a current through an external resistance R.

Answer (English): The total resistance of the circuit is R + r, so the current is E / (R + r). The potential drop across the external resistance is I R, which is the terminal voltage of the cell. Write E = I R + I r, so the terminal voltage is E - I r. When I is small, the terminal voltage is close to the emf; when the current is large, the lost volt I r becomes significant and the terminal voltage falls. If the cell is being charged, the current is forced opposite to its emf and the terminal voltage becomes E + I r.

Hinglish: Circuit ka total resistance R + r hai, isliye current E / (R + r) hota hai. External resistance par potential drop I R hi terminal voltage hai. E = I R + I r likhne par terminal voltage E - I r milta hai. Current chhota ho to terminal voltage emf ke kareeb hota hai, aur bada ho to I r ka loss dikhne lagta hai. Charging ke case mein current emf ke opposite hota hai, tab terminal voltage E + I r ho jata hai.

Q7. State Kirchhoff's junction rule and loop rule. Which conservation law does each express?

Answer (English): The junction rule states that the total current entering a junction equals the total current leaving it. It follows from conservation of charge, because charge cannot pile up at a junction in a steady state. The loop rule states that the algebraic sum of the potential differences around any closed loop is zero. It follows from conservation of energy, because a charge that travels around a closed loop returns to the same potential and cannot gain energy from nothing.

Hinglish: Junction rule kehta hai ki junction mein aane wala total current nikalne wale total current ke barabar hota hai. Yeh charge conservation se aata hai, kyunki steady state mein charge junction par jama nahi hota. Loop rule kehta hai ki kisi bhi closed loop ke around potential differences ka algebraic sum zero hota hai. Yeh energy conservation se aata hai, kyunki charge loop ghoom kar wapas usi potential par aata hai aur energy nahi bana sakta.

Q8. Explain the balance condition of a Wheatstone bridge.

Answer (English): Four resistances P, Q, R and S form the four arms of a bridge, with a galvanometer across one diagonal and a cell across the other. At balance the galvanometer shows no deflection, so no current flows through it and points on both ends of the galvanometer are at the same potential. Applying the loop rule to the two upper and lower paths gives the ratio condition P/Q = R/S. If the bridge is not balanced, current flows through the galvanometer and the potential difference across it is not zero.

Hinglish: Char resistances P, Q, R aur S bridge ke arms banate hain, galvanometer ek diagonal par aur cell doosri diagonal par hoti hai. Balance par galvanometer zero deflection deta hai, matlab usme current nahi behta aur dono sire same potential par hote hain. Loop rule lagane par ratio condition P/Q = R/S milti hai. Agar bridge balanced nahi hai to galvanometer mein current behta hai aur uske across potential difference zero nahi hota.

Q9. Explain how a potentiometer measures emf more accurately than a voltmeter.

Answer (English): A voltmeter connects across the cell and draws some current from it, so it measures the terminal voltage E - I r, not the emf. A potentiometer works by balancing the cell against a known potential drop along a uniform wire. At balance no current is drawn from the cell, so the terminal voltage equals the emf. The balance is a null method, so the galvanometer is used only as a zero detector and its calibration does not affect the result. That is why the potentiometer gives the true emf.

Hinglish: Voltmeter cell ke across lagta hai aur kuch current cell se le leta hai, isliye woh terminal voltage E - I r measure karta hai, emf nahi. Potentiometer uniform wire par known potential drop ke against cell ko balance karta hai. Balance par cell se current nahi liya jata, isliye terminal voltage emf ke barabar hota hai. Yeh null method hai, galvanometer sirf zero detect karta hai, isliye uski calibration result ko affect nahi karti. Isi wajah se potentiometer sahi emf deta hai.

Q10. Two identical cells of emf E and internal resistance r are connected first in series and then in parallel with a load R. Find the condition under which the current through R is the same in both cases.

Answer (English): In series the emfs add and the internal resistances add, so the current is 2E / (R + 2r). In parallel the equivalent emf is E and the equivalent internal resistance is r / 2, so the current is E / (R + r/2). Set the two currents equal: 2 / (R + 2r) = 1 / (R + r/2). Cross multiply to get 2R + r = R + 2r, which gives R = r. The currents are equal when the external resistance equals the internal resistance of one cell.

Hinglish: Series mein emf aur r dono add hote hain, isliye current 2E / (R + 2r). Parallel mein equivalent emf E aur equivalent internal resistance r/2 hota hai, isliye current E / (R + r/2). Dono ko barabar rakho: 2 / (R + 2r) = 1 / (R + r/2). Cross multiply karo to 2R + r = R + 2r, isliye R = r. Matlab jab external resistance ek cell ke internal resistance ke barabar ho, tab dono cases mein current same hota hai.

Quiz below: five questions on current density, drift speed, Ohm's law limits, cell terminal voltage and Wheatstone balance. Revise the formula sheet and the circuit rules page before you start.