The Physics
Hypertextbook
Opus in profectus

# Capacitors

## Problems

### practice

1. wax paper, aluminum foil capacitor
2. Write something.
3. The lowest part of Earth's atmosphere is called the troposphere. It is the part of the atmosphere where basically all weather happens. The prefix tropo- comes from the Greek wordτρόπος (tropos) meaning turn, a turnaround, or a change. The troposphere is that part of the atmosphere where all the day to day changes we call weather occur. One of the side effects of thunderstorms is the separation of charge in the troposphere — positive at the top, negative at the bottom. The atmosphere behaves like a circuit with properties like voltage, current, resistance, and capacitance. Given the title of this section, we'll be concentrating on the capacitance. To start this problem, you need to review some old physics related to fluids. To finish it, you need to apply some new physics related to electrostatics.
1. Determine the effective thickness of the troposphere — a.k.a. its scale height. What height would the atmosphere have if it was one uniform density from bottom to top? Hint: Use surface pressure to determine the mass of the atmosphere, assume the density on the surface is the density everywhere, and determine the thickness of this ideal atmosphere. Additional hint: Figure this out using symbols and algebra until you have one equation with thickness as the subject before you start playing with numbers.
2. Determine the total capacitance of the troposphere.
4. Write something completely different.

### conceptual

1. An air filled, parallel plate capacitor with capacitanceC is connected to a battery with voltageV resulting in a stored chargeQ. The battery remains connected and the separation between the plates is increased fromd to2d. Determine the new values of the following quantities in terms of their original values…
1. capacitance
2. voltage
3. charge
2. An air filled, parallel plate capacitor with capacitanceC is connected to a battery with voltageV resulting in a stored chargeQ. The battery is carefully disconnected and the separation between the plates is increased fromd to 2d. Determine the new values of the following quantities in terms of their original values…
1. capacitance
2. charge
3. voltage
3. How would the capacitance of a capacitor change if…
1. the charge on the capacitor was doubled?
2. the charge on the capacitor was halved?
3. the voltage across the capacitor was doubled?
4. the voltage across the capacitor was halved?

### numerical

1. One farad is a big capacitance. Determine the surface area of a 1 F air-filled capacitor with a plate separation of 1 mm.
2. A storm cell covers an area of one square kilometer. The clouds that make up the cell range in altitude from 1000 m to 9000 m. The clouds are charged negative on the bottom and positive on the top due to some incompletely understood mechanism (possibly friction between ice crystals and the air). The surface of the Earth acquires a positive charge by induction. (The negative charges are driven into the Earth.) Determine…
1. the direction of the electric field within the clouds of the storm cell
2. the direction of the electric field in the space between the bottom of the cell and the surface of the Earth
3. the capacitance of the storm cell
4. the capacitance of the space between the bottom of the cell and the surface of the Earth

The dielectric strength of air (the field value that would cause air to change from an insulator to a conductor) is 3,000,000 V/m. Determine…

1. the maximum voltage difference between the cloud and the ground that the air can sustain before lighting would strike
2. the maximum charge that can be stored in the Earth-cloud "capacitor"
3. Neurons (nerve cells) consist of three basic parts: the soma, which contains the nucleus; dendrites, which are short fibrous extensions that look like tree branches; and an axon, which is a long, slender extension on the side opposite the dendrites. The axon is the largest part of most neurons. A typical axon in the brain is about 1 cm long and 1 μm in diameter. In the resting phase, the outside is positively charged and the inside is negatively charged to a potential difference of −70 mV. The space between these charges is about 6 nm. Determine the following quantities for a typical axon in its resting phase…
1. its surface area
2. its capacitance (assuming it was an air-filled capacitor)
3. its stored charge
There are trillions of neurons in the human brain. Determine the following quantities for the human brain…
1. its effective total capacitance
2. the total charged stored in all of its neurons
4. Dynamic Random Access Memory (DRAM) chips are common components in computer systems. A gigabyte DRAM chip packs around eight billion capacitors in a square a few centimeters on a side. Each capacitor stores one bit of information. A charged capacitor represents a value of 1, while a discharged capacitor represents a value of 0. An often cited value in the semiconductor industry is that DRAM capacitors should have a minimum capacitance of 30 fF. An electrical engineer wishes to design a DRAM chip composed of 30 fF capacitors with a plate separation of 100 nm. (As a first approximation, assume the plates are separated with air.) Determine…
1. the area of one plate
2. the length of the side of one plate in nanometers