The given figure shows some of the given figure electric field lines due to three-point charges in the mentioned exercise. arranged along the vertical axis. All three charges have the same magnitude. (a) What are the signs of the three charges? Explain your reasoning. (b) At what point(s) is the magnitude of the electric field the smallest? Explain your reasoning. Explain how the fields produced by each individual point charge combine tn give a small net field at this point or points.

A dielectric with a dielectric constant $\kappa=4$ is inserted into a parallel plate capacitor, filling $\frac{1}{3}$ of the volume, as shown in the figure. If the capacitance of the capacitor without the dielectric is $C$, what is the capacitance of the capacitor with the dielectric? a) $0.75 C$ b) $C$ c) $2 C$ d) $4 C$ e) $6 C$

A parallel plate capacitor is attached to a battery that supplies a constant voltage. While the battery is still attached, a dielectric of dielectric constant $\kappa=3.0$ is inserted so that it just fits between the plates. What is the energy stored in the capacitor after the dielectric is inserted in terms of the energy $U_0$ before the dielectric was inserted?

Aluminum and copper wires of equal length are found to have the same resistance. What is the ratio of their radii?

A charge + Q is located at (0, 0, d), while another charge -Q is located at (0, 0, -d). Determine E: (a) at any point on the x-axis, (b) at any point on the z-axis.

How much charge is on each plate of a $4.00-\mu \mathrm{F}$ capacitor when it is connected to a 12.0-V battery?

(a) How much charge is on each plate of a 4.00-$\mu F$ capacitor when it is connected to a 12.0-V battery? (b) If this same capacitor is connected to a 1.50-V battery, what charge is stored?

(a) Sketch the electric field lines for an infinite line of charge. You may find it helpful to show the field lines in a plane containing the line of charge in one sketch and the field lines in a plane perpendicular to the line of charge in a second sketch. (b) Explain how your sketches show (i) that the magnitude $E$ of the electric field depends only on the distance $r$ from the line of charge and (ii) that $E$ decreases like $1 / r$.

Sketch the electric field lines for a disk of radius $R$ with a positive uniform surface charge density $\sigma$. Use what you know about the electric field very close to the disk and very far from the disk to make your sketch.

The separation between two enormous parallel sheets of charge is $d$. The surface charge densities of the two sheets are positive $\sigma>0$ for one and negative  $-\sigma<0$ for the other. Draw the electric field lines in locations that are far from the sheets' edges and close to the sheet's centre.

Suppose the charge shown in the given figure is fixed in position. A small, positively charged particle is then placed at some point in the figure and released. Will the trajectory of the particle follow an electric field line? Why or why not? Suppose instead that the particle is placed at some point in the given figure and released (the positive and negative charges shown in the figure are fixed in position). Will its trajectory follow an electric field line? Again, why or why not? Explain any differences between your answers for the two different situations.

An infinite planar sheet of paper has surplus electrons that are  $2.50 \times 10^6$ per square centimetre. Determine the strength and direction of the electric field at a point that is $5.00 \mathrm{~cm}$ from the sheet's surface. If the sheet is big enough, it can be thought of as an infinite plane.

Two horizontal, infinite, plane sheets of charge are separated by a distance d. The lower sheet has a negative charge with uniform surface charge density $-\sigma<0$. The upper sheet has a positive charge with uniform surface charge density $\sigma>0$. What is the electric field (magnitude, and direction if the field is nonzero) (a) above the upper sheet, (b) below the lower sheet, (c) between the sheets?

Infinite sheet $A$ carries a positive uniform charge density $\boldsymbol{\sigma}$, and sheet $B$, which is to the right of $A$ and parallel to it, carries a uniform negative charge density $-2 \sigma$. (a) Sketch the electric field lines for this pair of sheets. Include the region between the sheets as well as the regions to the left of $A$ and to the right of $B$. (b) Repeat part (a) for the case in which sheet $B$ carries a charge density of $+2 \sigma$.