PHYS 2015 -- Week 6 Sharpen thinking about connections among electric field, electric potential difference, potential energy Apply the ideas to capacitance and the parallel plate capacitor For exclusive use in PHYS 2015. Not for re-distribution. Some materials Copyright University of Colorado, Cengage,, Pearson J. Maps.
x f i x y x x y z z z Electric Field as slope (or gradient) of V dx v v E The electric field points toward lower V. The electric field tells us the steepness of the hill (slope) V(x) volts / meter = newtons / coulomb x V dv E E d r dx E dv dx dv E E d r ( E V E x dx V y E E dy E V dz)
y Cross-sections of equipotential surfaces (equipotential lines) are shown for three uniform electric fields between charged planes. All three cases are for the same size region of space.
y Equipotential lines are shown for three uniform electric fields. In which case(s) is the electric field directed up? 1. (1) 2. (2) 3. (3) 4. (1) and (2) 5. (2) and (3) 20% 20% 20% 20% 20% 10 1. 2. 3. 4. 5.
y Equipotential lines are shown for three uniform electric fields. Which case has the largest electric field (in magnitude)? 1. (1) 2. (2) 3. (3) 4. (2) and (3) 5. All the same 0% 0% 0% 0% 0% 10 1. 2. 3. 4. 5.
y Equipotential lines are shown for three uniform electric fields. The distance from top to bottom is 0.50 m. The electric field for case 1 is 1. 80 V/m 2. 40 V/m 3. 160 V/m 4. 240 V/m 5. Impossible to say 0% 0% 0% 0% 0% 10 1. 2. 3. 4. 5.
The electric potential V(x) as a function of position is shown. Which region has the most positive electric field component, E x? 1. 1 2. 2 3. 3 4. 4 5. 5 0% 0% 0% 0% 0% 10 1. 2. 3. 4. 5.
Capacitance and Capacitors, Batteries Batteries (ideally) proved a constant electric potential difference ΔV or E between their terminals. They can do work on charges moving them from one terminal to the other and thereby give the charges electric potential energy U = q ΔV. This energy comes from chemical energy in the battery. Capacitors are devices for storing charge and electric potential energy. The energy can be viewed as stored in the electric fields of the charges. Our standard view of a capacitor will be two nearby conductors carrying equal but opposite charges. The most important geometry will be parallel plates. We can also have cylindrical capacitors (commonly seen as coaxial cables) and spherical capacitors
Charging a parallel plate capacitor with a battery Battery s job is to maintain a fixed potential difference ()V Conventional current thinking = Motion of positive charges Conventional View: Battery pushes + charges onto upper plate from its positive terminal, repelling positive charges from lower plate. These + charges are taken in by the negative battery terminal. Microscopic Reality: Battery s + terminal attracts electrons from the top plate, leaving it positively charged, and that attracts electrons onto the lower plate from the battery s negative terminal.
Parallel Plate Capacitor ΔV +Q +Q -Q -Q
Energy Storage in Capacitors The work done charging a capacitor is stored as potential energy U, which may be viewed as stored in the electric field between the capacitor plates. Move some charge dq from 1 plate to the other, then another dq, then another. As V(q ) builds up it takes more work to move each dq Initially uncharged Intermediate amount of charge q, potential difference V Fully charged, final Q, V
When the voltage or potential difference across a capacitor is doubled (by charging it with two batteries in series instead of just one), the capacitance of the capacitor is 1. Doubled 2. Unchanged 3. Cut in half 0% 0% 0% 10 1. 2. 3.
Worksheet Work parts 1 and 2 Think about 3 and 4 in combination answering 4 first may help you answer 3
A parallel plate capacitor is charged and the plates are isolated so Q cannot change. The plates are then pulled apart so that the plate separation d increases. The electric field between the plates of the capacitor. 1. Decreases 2. Remains constant 3. Increases d +++++++++++++++++++++++ E ----------------------------------------- +Q -Q 0% 0% 0% 10 1 2 3
A parallel plate capacitor is charged and the plates are isolated so Q cannot change. The plates are then pulled apart so that the plate separation d increases. The capacitance of the capacitor. 1. Decreases 2. Remains constant 3. Increases d +++++++++++++++++++++++ E ----------------------------------------- +Q -Q 0% 0% 0% 10 1 2 3
PHYS 2015 -- Week 6 Reading Journals Thursday Capacitors in combination, Dielectrics, Electric current So far: Q = CV U = (1/2) CV 2 C = ε o A / d (remember V really is the potential difference ΔV = Ed across the capacitor) For exclusive use in PHYS 2015. Not for re-distribution. Some materials Copyright University of Colorado, Cengage,, Pearson J. Maps.
A parallel-plate capacitor has square plates of edge length L, separated by a distance d. If we double each dimension L and halve the dimension d, by what factor have we changed the capacitance? 1. Decreased to (1/2) x original 2. No change 3. Increased to 2x original 4. Increased to 4x original 5. Increased to 8x original d L 96% Adapted from University of Colorado Boulder 4% 0% 0% 0% 1. 2. 3. 4. 5.
A parallel plate capacitor is connected to a battery and charged. While still connected to the battery, the plates are then pulled apart so that the plate separation d increases. The total electrostatic energy stored in the capacitor. 1. Decreases 2. Remains constant 3. Increases E d +++++++++++++++++++++++ E ----------------------------------------- 0% 0% 0% 1 2 3
Capacitors in Parallel share the same electric potential difference (voltage) ΔV Replaceable by a single equivalent capacitor C eq = C 1 + C 2
Capacitors in Series store the same amount of electric charge, Q 1 = Q 2 = Q Replaceable by a single equivalent capacitor 1 C eq 1 C 1 1 C 2
Two capacitors C 1 and C 2 are connected as shown (in series). Then I connect a battery so V(ac) = 6 V. What are V(ab) and V(bc)? 1. Both are 6 V 2. Both are 3 V 3. V(ab)=4 V, and V(bc) = 2 V 4. V(ab)=2 V, and V(bc) = 4 V a C1 =2 mf b C2 =4 mf c 100% 0% 0% 0% Adapted from University of Colorado Boulder 1. 2. 3. 4.
What is the effective total capacitance of this set of (ridiculously large) capacitors? C 2 =2 F 1. 8 F 2. 5 F 3. 4 F 4. 2 F 5. Other C 1 =4 F C 3 =2 F 75% 18% 7% 0% 0% 1. 2. 3. 4. 5.
Quick: Find C eq for each
What Q total does a 3V battery supply when connected to the terminal points
MEMS Airbag Sensor (sensorsmag.com)
Energy Density (u= U/Volume ) in the space between plates Stored Potential Energy U 3 useful forms via Q=CV QV C Q CV U 2 1 2 1 2 1 2 2 2 0 2 0 2 0 2 0 2 2 1 2 1 2 1 ) ( 2 1 2 1 E u E Ad U Ad E Ed d A CV U E
An insulator placed between the plates of a capacitor to increase the capacitance is referred to as a 1. Polarizer 2. Polar enhancer 3. Dielectric 4. Diuretic 5. Dipole moment 0% 0% 0% 0% 1 2 3 4
An insulator placed between the plates of a capacitor to increase the capacitance is referred to as a 1. Polarizer 2. Polar enhancer 3. Dielectric 4. Diuretic 5. Dipole moment 0% 0% 0% 0% 1 2 3 4
Capacitors with dielectric materials Polar dielectrics polar molecules, align with the electric field Non-polar dielectrics molecules stretch and become polarized, For fixed Q, (charge plates then battery removed) E and V decrease with dielectric added, E = E 0 / κ V = V 0 / κ For fixed V, (battery remains connected) Q increases with dielectric added (with the extra charge delivered from battery) C C 0
A parallel plate capacitor is charged and the plates are isolated so Q cannot change. A slab of insulator is then inserted between the plates. Upon insertion of the slab, the voltage difference between the plates 1. Decreases 2. Remains the same 3. increases +++++++++++++++++++++++ +Q d -Q ----------------------------------------- 96% 4% 0% 1. 2. 3.
Two parallel plates of area 100 cm^2 are given charges of equal magnitudes of q=0.89 μc, but opposite signs. When a dielectric material is inserted filling the space, the net electric field between the plates is 1.4 x 10^6 V/m. Calculate the dielectric constant of the material.