Questions: Electric Charges and Fields

Q1. What does quantisation of electric charge mean, and why can we ignore it for a large body?

Answer (English): Quantisation means every charge is a whole-number multiple of the elementary charge e, which is 1.6 × 10^-19 C. A body with charge 2.4 × 10^-19 C is impossible because the number of elementary charges cannot be 1.5. For a body of ordinary size the total charge is built from an enormous number of electrons, so removing a few million electrons changes the charge by a fraction far too small to notice. In that limit the charge behaves like a continuous quantity and the discrete steps are ignored.

Hinglish: Quantisation ka matlab hai charge hamesha e ke poore multiple mein hota hai, jahan e = 1.6 × 10^-19 C hai. Aadha electron ya do se dhai electron possible nahi hai. Bade body par electrons ki sankhya bahut zyada hoti hai, isliye kuch million electrons nikalne se charge mein itna chhota badlaav aata hai ki notice nahi hota. Isliye macroscopic problems mein charge ko continuous maan lete hain.

Q2. State Coulomb's law in vector form and explain why the force in a medium is smaller than in vacuum.

Answer (English): For two point charges q1 and q2 separated by a vector r12 that points from charge 2 to charge 1, the force on charge 1 is k q1 q2 divided by r12 squared, multiplied by the unit vector along r12. The force on charge 2 is the negative of this. The medium weakens the force because its molecules polarise and set up their own field, which partly cancels the field of the two charges. The dielectric constant K measures this reduction, and the force in the medium is the vacuum force divided by K.

Hinglish: Vector form mein F12 = k q1 q2 r12 hat / r12^2, aur doosre charge par force iska ulta hota hai. Medium mein force kam hota hai kyunki medium ke molecules polarise ho jate hain aur apni field bana kar original field ko thoda cancel karte hain. Dielectric constant K yeh batata hai ki force kitna guna kam hua, aur medium ka force vacuum ke force ko K se divide karke milta hai.

Q3. Why do electric field lines never cross, and why can they not form closed loops in electrostatics?

Answer (English): The tangent to a field line gives the direction of the field at that point. If two lines crossed, the field would have two directions at the same point, and a vector cannot point two ways at once. Closed loops are impossible because the electrostatic force is conservative, so the work done in carrying a test charge around any closed path is zero. A closed field line would mean the field does positive work around a loop, which contradicts the conservative property.

Hinglish: Field line ka tangent us point par field ki direction deta hai. Agar do lines cross karein to ek hi point par field ki do direction ho jayengi, jo possible nahi hai. Closed loop isliye possible nahi hai ki electrostatic force conservative hai, isliye kisi bhi closed path par work zero hota hai. Agar field line band hoti to field ka work nonzero hoga, jo contradiction hai.

Q4. Define electric flux. Why can it be negative, and what is its SI unit?

Answer (English): Electric flux through a surface is the dot product of the electric field and the area vector of the surface. It measures how much field pierces the surface. For a closed surface the outward normal is taken as positive, so field leaving the surface gives positive flux and field entering gives negative flux. The dot product is negative when the angle between the field and the area vector is more than 90 degrees. The SI unit is N m^2 C^-1.

Hinglish: Electric flux surface ke through field kitna guzarta hai, wo batata hai, aur formula E S cos theta hai. Closed surface ke liye bahar ki normal positive hoti hai, isliye field bahar nikle to positive flux aur andar aaye to negative flux. Jab field aur area vector ke beech ka angle 90 degree se zyada ho to dot product negative aata hai. Iski SI unit N m^2 C^-1 hai.

Q5. Define the electric dipole moment and compare the axial and equatorial fields of a dipole at the same large distance.

Answer (English): The dipole moment of a pair of charges q and -q separated by 2a has size p = q × 2a, and its direction runs from the negative charge to the positive charge. At a far point on the axis the field is 2k p / r^3 and points along the dipole moment. At a far point on the equatorial plane the field is k p / r^3 and points opposite to the moment. The axial field is therefore twice the equatorial field at the same distance, and both fall as the cube of the distance rather than its square.

Hinglish: Dipole moment ka size p = q × 2a hota hai aur direction negative charge se positive charge ki taraf hoti hai. Axis ke far point par field 2kp / r^3 hoti hai aur dipole moment ke along hoti hai. Equatorial plane ke far point par field kp / r^3 hoti hai aur moment ke opposite hoti hai. Isliye axial field equatorial field ka double hota hai, aur dono r ke cube ke saath ghatte hain, square ke saath nahi.

Q6. Derive the torque on an electric dipole in a uniform electric field.

Answer (English): Let the dipole moment make an angle theta with the field. The forces on the two charges have size qE each and point in opposite directions, so the net force is zero and the dipole does not translate. The two forces form a couple. Take the centre of the dipole as the axis; each force has a perpendicular distance a sin theta from it, so each contributes qE a sin theta, and the two add to 2qE a sin theta. Since p = q × 2a, the torque is p E sin theta. In vector form it is the cross product of p and E.

Hinglish: Dipole moment field ke saath theta angle banata hai. Dono charges par force qE hoti hai aur opposite direction mein hoti hai, isliye net force zero hota hai aur dipole move nahi karta, sirf ghoomta hai. Dono forces milkar couple banati hain. Har force ka perpendicular distance centre se a sin theta hota hai, isliye dono ka torque qE a sin theta aur total 2qE a sin theta hota hai. p = q × 2a rakhne par torque p E sin theta milta hai, aur vector form mein yeh p aur E ka cross product hai.

Q7. State Gauss's law and explain why the flux through a closed surface does not depend on the shape of the surface.

Answer (English): Gauss's law says the total electric flux through any closed surface equals the net charge enclosed divided by epsilon0. The reason the shape does not matter is geometric. For a point charge at the centre of a sphere, the flux is the field times the area, and the radius cancels. Any closed surface around the same charge can be split into pieces of spheres and cones, and each piece contributes the same solid angle, so the total again depends only on the charge. A charge outside the surface contributes zero net flux because every field line that enters also leaves.

Hinglish: Gauss's law kehta hai kisi bhi closed surface ka total flux andar ke net charge ka 1 / epsilon nought guna hota hai. Surface ki shape se farq nahi padta kyunki geometry cancel ho jati hai. Point charge ko ghere sphere mein field area se multiply karo, radius cancel. Koi bhi closed surface ko chhote sphere aur cone ke tukdon mein baant sakte hain, har tukda same solid angle deta hai, isliye total sirf charge par depend karta hai. Bahar ka charge zero net flux deta hai kyunki jitni field lines andar jaati hain utni hi bahar aati hain.

Q8. Use Gauss's law to find the electric field due to an infinite plane sheet of charge.

Answer (English): Let the sheet have uniform surface charge density sigma. Choose a cylindrical pillbox that crosses the sheet with its flat faces parallel to the sheet. By symmetry the field points away from the sheet on both sides, so it is parallel to the curved surface and produces no flux there. Each flat face has area A and carries flux E A. The net flux is 2 E A. The charge enclosed is sigma A. Gauss's law gives 2 E A = sigma A / epsilon0, so E = sigma / (2 epsilon0). The result is independent of the distance from the sheet.

Hinglish: Sheet ki surface charge density sigma lo. Ek pillbox choose karo jo sheet ko cross kare aur dono flat faces sheet ke parallel hon. Symmetry se field sheet se dono taraf bahar hoti hai, isliye curved surface ke parallel hoti hai aur wahan flux zero hota hai. Dono flat faces ka flux E A aur E A hai, total 2 E A. Andar ka charge sigma A. Gauss's law se 2 E A = sigma A / epsilon nought, isliye E = sigma / (2 epsilon nought). Yeh answer sheet se distance par depend nahi karta.

Q9. A charged conductor has no field inside its material. Explain using Gauss's law.

Answer (English): In a conductor at rest the charges have already moved until the field everywhere inside the material is zero. Draw a small Gaussian surface completely inside the conductor. The field is zero on every point of that surface, so the flux is zero, and Gauss's law then says the enclosed charge is zero. This holds for every such surface, so there is no net charge anywhere in the interior. Any excess charge must sit on the outer surface.

Hinglish: Conductor ke andar charges tab tak move karte hain jab tak field zero na ho jaye. Conductor ke andar ek chhota Gaussian surface lo. Us surface ke har point par field zero hai, isliye flux zero, aur Gauss's law kehta hai ki andar ka charge zero hona chahiye. Yeh har aisi surface par sach hai, isliye interior mein charge nahi hota. Zyada charge conductor ke bahar ki surface par baith jata hai.

Q10. Explain why a field line cannot have a sudden break.

Answer (English): A field line has no beginning or end except on a charge. If a line broke in empty space, the field would have to jump from one value to another across that gap, which would mean an infinite rate of change of the field there. The field of any real charge distribution varies continuously in empty space, so a line must continue smoothly until it reaches a charge, leaves the region, or goes to infinity. A break is possible only where a charge sits, because a charge is a source or sink of field lines.

Hinglish: Field line ka start ya end sirf charge par hota hai. Agar line khaali space mein toot jaye to us gap par field ek value se doosri value par jump karegi, jo possible nahi hai. Real charge distribution ki field empty space mein continuous hoti hai, isliye line smooth chalti rehti hai jab tak charge na mile ya infinity tak na chali jaye. Break sirf charge ke paas ho sakta hai kyunki charge field lines ka source ya sink hota hai.

Quiz below: five questions on charge properties, Coulomb's law, the field of a point charge, flux and Gauss's law. Revise the formula sheet and the Gauss applications page before you start, then check any wrong answer against the topic pages.