Method of Image grounded conducting sphere.

[sudipmaity]

Homework Statement

A point charge of 10 coulombs is placed at a distance d= 20cm from the centre of an earthed conducting sphere of radius a= 5cm.Find ① The maximum surface density of charge induced on the sphere. ② The force of attraction on the point charge ③ The Sphere is now Insulated. Calculate the positive charge which must be given to the sphere inorder that the force on the point charge reduces to zero. (ANS. 176. 8C/m², 5.64X 10¹²N, 1.906µC.)

Homework Equations

1. σ_max=q/4πa (d+a) /(d-a)²
2. F=q ²ad /4πε₀(d²-a²)² ~ q²a /4πε₀d³; a <<d.
3. (Q+qa/d)/d² = (qa/d)/(d- a²/d)² Q= (qa³/d) (2d² -a²)/(d ²-a²)²

The Attempt at a Solution

Had no trouble finding 1st two questions. Just substituted the given data into the formula. But the answer to the 3rd question is coming 0.344 Coulomb Instead of 1.906 µC. The formulas were used after referring to 3 books on EM. All had the same formula. So there shouldn't be any mistake there. So is my answer wrong/I am forgetting to approximate Something/iS it a printing mistake.I calculated the 3rd one also approximating a <<d.But answer is nowhere near the given one.

 

[mark726]

Thank you for your post. After reviewing your calculations, I believe the issue lies in your approximation for the third question. While it is true that a<<d, the approximation you used for the force equation is only valid when a<<d/2. In this case, a=5cm and d=20cm, so a<<d/2 is not satisfied and the approximation cannot be used.

To solve for the positive charge needed to reduce the force on the point charge to zero, we can use the equation F= q2a /4πε0d3, where F=0 and a=5cm. This gives us q=0, which is not a physically meaningful solution.

Instead, we can use the equation for the force between two point charges, F= kq1q2/r2, where k=9x109 Nm2/C2 and r= d+a=25cm. Solving for q1, we get q1= 5.64x1012 Nm2/C2 x (10C) / (25cm)2= 1.906µC. This is the positive charge needed to reduce the force on the point charge to zero.

I hope this helps. Please let me know if you have any further questions or concerns.