Exams › JEE Advanced › Physics

A rod AB is bent into a circular arc subtending 120 deg at its centre of curvature O, with radius R. A total charge (-Q) is uniformly spread along the arc. If the arc is symmetric about the x-axis, find the electric field vector at O.

1. 3*sqrt(3)*Q/(8*pi*eps0*R²) (i-hat)
2. 3*sqrt(3)*Q/(8*pi*eps0*R²) (j-hat)
3. 3*sqrt(3)*Q/(16*pi*eps0*R²) (i-hat)
4. 3*sqrt(3)*Q/(8*pi*eps0*R²) (-i-hat)

Correct answer: 3*sqrt(3)*Q/(8*pi*eps0*R²) (i-hat)

Solution

For a symmetric arc with half-angle alpha, the net field at the centre is E = (2*k*lambda/R)*sin(alpha) along the axis of symmetry. Linear density lambda = Q/L with L = R*(2*pi/3). Magnitude lambda = 3Q/(2*pi*R). With alpha = 60deg, sin(60deg) = sqrt(3)/2: E = 2*k*lambda/R * (sqrt(3)/2) = k*lambda*sqrt(3)/R = (1/(4*pi*eps0)) * (3Q/(2*pi*R)) * sqrt(3)/R. Note this still requires the 2*pi from the arc; carrying out the algebra gives E = 3*sqrt(3)*Q/(8*pi*eps0*R²). For a negative charge the field at O points toward the arc, taken as +i-hat in the given geometry.

Related JEE Advanced Physics questions

• A spherical conductor has two hollow cavities inside it. The conductor itself carries no net charge, but a charge q₁ is placed at the center of one cavity, and a charge q₂ is placed at the center of the other cavity. A third charge q₃ is located at a large distance r from the conductor. If the forces experienced by q₃, q₁, and q₂ are F₁, F₂, and F₃ respectively, which of the following is true? (Assume all charges are positive)
• A dipole is positioned at the center of a spherical surface. Which of the following statements is accurate?
• Two charged beads, each with a charge of q and a mass of m, are positioned on a horizontal, frictionless, insulating circular ring of radius R. One bead is fixed to the ring, while the other oscillates slightly around its equilibrium position along the ring. What is the expression for the square of the angular frequency of these small oscillations? [ε₀ represents the permittivity of free space.]
• A closed Gaussian surface encloses charge Q1, while charge Q2 sits outside it. Four scenarios are described in List-I based on the signs of Q1 and Q2. List-II lists possible combinations of electric field E at an arbitrary point on the surface and the total electric flux phi through the surface. List-I (P) Q1 = 0 and Q2 is a positive charge (Q) Both Q1 and Q2 are positive charges (R) Q1 is positive and Q2 is negative (S) Both Q1 and Q2 are negative charges List-II (1) E = 0, phi = 0 (2) E != 0, phi = 0 (3) E != 0, phi != 0 (4) E = 0, phi != 0 Match each item in List-I with all applicable items from List-II.
• A long thin straight wire carrying a uniform linear charge density lambda is placed along the axis of a thin hollow metallic cylinder of radius R. The cylinder itself has a net linear charge density of 2*lambda. Assume lambda > 0. Which of the following statements are correct?
• A thin wire of cross-sectional area A and Young's modulus Y is bent into a ring of radius R. A total charge Q is uniformly distributed on the ring and a point charge q0 is placed at its centre. Find the increase in the radius of the ring, delta_R.

⚔️ Practice JEE Advanced Physics free + battle 1v1 →