# Gauss' Law Practice Problems

Along the axis of an infinite cylindrical conducting shell runs a long line of charges with a net uniform linear charge density of 4 C/m. The shell has an inner radius R_{1} and an outer radius R_{2 }and a uniform linear density of 4 C/m. Find the charge per unit length on the i) inner and ii) outer surfaces of the shell.

A straight, infinitely long electrical line wire of positive charge per unit length +λ, coincides with the axis of an infinite cylindrical conducting pipe. The pipe has inner and outer radii of R_{1} and R_{2}, respectively, and carries a positive charge per unit length λ. λ is a positive constant expressed in units of C/m. Determine the electric field expression as a function of λ and r, the distance from the pipe's axis i) r < R_{1} ii) R_{1 }< r < R_{2} iii) r > R_{2}.

A very long metallic rod of radius R carries a uniform charge per unit area σ. Determine the expression for the magnitude of the electric field induced by the charged rod at a distance r > R from its axis as a function of σ and the linear charge density λ.

Determine the linear density of a very long cylindrical rod in terms of the charge per unit area σ and the radius R of the rod.

A particle with a mass of m_{1} = 8.14 × 10^{-6} kg and a charge of q_{1} = 2.83 × 10^{-11} C is at rest 5.00 cm above a large flat non-conducting plate. The plate has a constant charge density of 5.0 × 10^{-5} C/m. What must be the charge q_{2} of a particle of mass m_{2} = m_{1} for it to remain stationary when placed without initial speed at 2.00 cm above the plate?

A tiny drop of oil of mass 1.5 × 10^{-5} g and charge q is released from rest above an infinite non-conducting plate. The plate has a uniform charge density of 10^{-5} C/m. Determine the charge q if the oil drop remains stationary when it is released.

A point charge of -10 nC is located within a spherical metal shell that carries a total charge of 30 nC. What is the net charge on the i) internal (q_{int}) and ii) external (q_{ext}) surfaces of the shell?

A small ball with a net uniform positive charge of 50 nC and a radius of r = 0.2 cm is surrounded by a concentric thin spherical shell with a uniform negative charge of -50 nC and a radius of R = 1 cm. Find the net electric field (E) produced by the shell at the ball surface.

The helium atom consists of a spherical nucleus containing 2 protons. The nucleus is surrounded by 2 electrons orbiting around it at a distance of 1.0 × 10^{−}^{12} m. The radius of the helium nucleus is 1.6 × 10^{−15} m. Find the strength of the electric field generated by the nucleus at the electrons' orbit.

The earth is approximately a sphere of radius 6.37 × 10^{6} m. Together with its atmosphere, it is electrically neutral. The Earth carries a negative electric charge on its surface, Q_{Earth} = -6.8 × 10^{5} C. Calculate the magnitude of electric field (E) produced by the Earth at a location near its surface.

A uniform glass globe rubbed with silk acquires a positive charge, Q. The charge is distributed evenly across the globe's volume. The globe has a radius of 15.0 cm. The strength of the electric field just outside the globe is 350.0 N/C. Calculate the electric field at 10.0 cm from the globe center.

A solid ball made of polystyrene, an insulating material, has a negative charge of q spread uniformly over its volume. The ball has a radius of 6.00 cm. The magnitude of the electric field measured 10.00 cm from the center of the ball is 245 N/C. Find the volume charge density for the insulating ball.

A positive point charge q_{1} = 2.5 μC is fixed at the center of charged thin conducting spherical shell. The charge density of the shell is 85.0 μC/cm2. The internal and external radii of the shell are r_{1 }= 12.0 cm and r_{2 }= 12.5 cm, respectively. Determine the total electric flux (Φ_{E}) through a sphere centered at the point charge q_{1} and having radius R, where R < r_{1}.

A positive point charge of -5 μC is held at the center of a thin, hollow spherical shell. The shell has an internal radius of 7 cm, an external radius of 8 cm, and an initial surface charge density of 20 nC/cm^{2}. What is the magnitude of the electric field (E) near the surface of the shell?

A -15.00 μC point charge is placed at the center of a hollow, metallic spherical shell. The shell has an internal radius of 34.00 cm and an external radius of 35.00 cm. Initially, the hollow shell carries a charge density of 18.00 μC/m^{2} uniformly spread over its surface. Calculate the new surface charge density (σ_{new}) of the shell.

The electric flux measured at the surface of a spherical dust particle surface is -1.4 × 10^{2} N•m^{2}/C. The radius of the particle is 10 μm. What is the charge density (σ) on the surface of the particle? Assume that the net charge is spread uniformly throughout the particle's surface.

Suppose that a certain spherical planet of radius 3200 km generates an electric flux of -2.14 × 10^{18} N•m^{2}/C near its surface. Find i) the magnitude and ii) the direction of the electric field near the surface of the planet.

A spherical asteroid produces a net electric flux of 5.26 × 10^{6} N•m^{2}/C at its surface. Find the net charge (q_{net}) of the asteroid.

A metal spherical ball of radius 5 cm carries a charge of 50 nC. Determine the electric field (E) at a distance of 0.01 cm from the outer surface of the ball.

A conducting solid sphere of radius 10 cm is charged by conduction with a positive rod. The sphere acquires a charge of 10 μC. Determine the electric field, E, at point A located just inside the sphere.

A student measures the strength of the electric field created by a small positively charged object at a distance of R = 0.2 m from it and comes up with a value of 2 × 10^{6} N/C. Find i) the electric flux (Φ_{E}) through a Gaussian sphere of radius R centered at the location of the object and ii) the charge (q) of the object.