Lecture 6: Electric Charges and Field

Introduction

• Topic: Finding electric field on the axis of a charged ring
• Importance: Commonly asked in board exams and competitions.
• Lecturer: Alok Pandey

Overview of a Charged Ring

• Charged Ring: Continuous charge distribution.
• Electric field calculation is based on charge distribution rather than point charges.

• For a point charge, electric field is calculated as:

  [ E = \frac{kQ}{x^2} ]

Electric Field on the Axis of a Charged Ring

• Consider a ring with total charge Q uniformly distributed.
• We need to find the electric field at a point along the axis at a distance x from the center of the ring.

Electric Field Calculation Steps

1. Charge Element:
   • Consider a small element of charge dq.
   • Calculate electric field dE due to dq.

   • Use the formula for point charge:

     [ dE = \frac{k dq}{r^2} ]
   • Components of dE:
     • dE cos(θ) contributes to the electric field along the axis.
     • dE sin(θ) cancels out due to symmetry.
2. Integration:
   • Total electric field is the sum of all dE cos(θ) components.

   • Use integration over the entire ring:

     [ E = \int dE cos(θ) = \frac{kQx}{(R^2 + x^2)^{3/2}} ]
3. Final Expression for Electric Field:

   • At a distance x from the center of the ring:

     [ E = \frac{kQx}{(R^2 + x^2)^{3/2}} ]
     • Where R is the radius of the ring.

Special Cases

• At the Center of the Ring:
  • When x = 0, E = 0 (due to symmetry).
• At Infinity:
  • As x approaches infinity, E approaches 0.

Graph of Electric Field (E) vs Distance (x)

• The graph shows E is 0 at both the center and at infinity, with a maximum value in between.
• Electric field is positive on one side of the ring and negative on the other due to the nature of vector quantities.

Finding Maximum Electric Field

• To find the value of x that maximizes E:
  • Derive the expression for E with respect to x and set it to zero.
• Resulting value of x for maximum E:
  • x = ±R / √2

• Maximum Electric Field Value:

  [ E_{max} = \frac{2kQ}{3\sqrt{3}R^2} ]_

Conclusion

• Understanding the electric field on the axis of a charged ring is essential for both board exams and competitive examinations.
• Reminder to keep studying and practicing.