Why are Gaussian surfaces chosen such that the electric field is perpendicular or parallel to the surface?

A parallel-plate capacitor with plate area A and separation d contains a dielectric with constant κ and thickness d/2 adjacent to one plate. The equivalent capacitance is:

A superconducting quantum interference device has three superconducting wires meeting at a junction. If the current density in wire 1 is $$j_1 = 2.5\times10^6~\mathrm{A/m^2}$$ (cross-sectional area $$A_1 = 2.0\times10^{-8}~\mathrm{m^2}$$) and in wire 2 is $$j_2 = 1.8\times10^6~\mathrm{A/m^2}$$ (area $$A_2 = 3.0\times10^{-8}~\mathrm{m^2}$$), what is the current density in wire 3 (area $$A_3 = 4.0\times10^{-8}~\mathrm{m^2}$$) if all currents flow away from the junction?

A resistor dissipates power P when connected to a voltage source V. If both the voltage and the resistance are doubled, what is the new power dissipation?

A circular loop of wire with radius R carries a current I. If the radius of the loop is doubled while the current remains constant, by what factor does the magnetic field at the center of the loop change?

A cylindrical copper wire with a radius of 0.5 mm carries a current of 2.0 A. If the current density is uniform throughout the wire, what is the magnitude of the current density?

Which of the following best describes the energy transformation in an LR circuit?

A battery uses a chemical reaction with standard Gibbs free energy change ΔG° = -193 kJ/mol. If two electrons are transferred per reaction, what is the theoretical emf of this battery?

An electron with initial kinetic energy of 2.5 eV enters a region with a uniform electric field. If the electron gains 4.0 eV of potential energy as it moves through this region, what is its final kinetic energy?

Two magnetic field vectors $$\vec{B}_1 = 3\hat{i} - 4\hat{j} + 2\hat{k}$$ and $$\vec{B}_2 = -2\hat{i} + 5\hat{j} - 3\hat{k}$$ (in tesla) exist at a point. What is the angle between these two vectors?

If the distance `r` between two charges is doubled, how does the electric potential energy `U` change?

A square wire loop with side length 0.2 m and resistance 5 Ω is pulled at a constant speed of 3 m/s perpendicular to a uniform magnetic field of 0.5 T. The magnetic field is perpendicular to the plane of the loop. What is the magnitude of the induced current in the loop at the instant when exactly half of its area remains inside the magnetic field region?

Which method can change the charge distribution without direct contact?

What does the variable `r` represent in the equation $$U = \frac{kq_1q_2}{r}$$?

A parallel plate capacitor with air between its plates has capacitance $$C_0$$. When a dielectric with constant $$\kappa$$ is inserted to fill exactly half the space between the plates (so that one half, parallel to the plates, is filled with dielectric and the other half is just air), what is the new capacitance?

Three identical small charged spheres with charge q are placed at the vertices of an equilateral triangle with side length a. If one sphere is moved to the center of the triangle, by what factor does the magnitude of the net electric force on this sphere change?

Two identical conducting spheres are initially neutral. Sphere A is brought near a negatively charged rod without touching. While the rod remains close to Sphere A, the two spheres are connected with a conducting wire, then the wire is removed, and finally the rod is taken away. What is the final charge state of the spheres?

How is the time constant of an LR circuit calculated?

What is the primary characteristic of the electric field between two charged parallel plates in a capacitor?

Why are straight wires typically modeled as having zero inductance?