MCAT Physics and Math Review
Chapter 6: Circuits
Conclusion
This chapter covered a lot of material. We began with a review of current, taking special note of the conventional definition of current as the movement of positive charge (when, in fact, negatively-charged electrons are actually moving). We considered the basic laws of electricity and circuits: Kirchhoff’s laws, which are expressions of conservation of charge and energy, and Ohm’s law, which relates voltage, current, and resistance. We defined resistance and analyzed the relationships between resistance and resistivity (directly proportional), resistance and length (directly proportional), and resistance and cross-sectional area (inversely proportional). We also defined capacitance as the ability to store charge at some voltage, thereby storing energy. Throughout, we stressed the importance of the both the conceptual and mathematical treatment of resistors and capacitors in series and in parallel as a major testing topic on the MCAT. Finally, we covered the different meters that can be used to measure circuit quantities.
Electricity is often a challenging concept for MCAT students. Unlike kinematics, thermodynamics, and fluids, which are often more tangible, electricity is often best understood through schematics and models. Take time to review these last two chapters, as they will assuredly pay off as points on Test Day. In the next chapter, we turn our attention to a completely different topic that is no more tangible—but is far more audible: sound.
Concept Summary
Current
· Current is the movement of charge that occurs between two points that have different electrical potentials.
o By convention, current is defined as the movement of positive charge from the high-potential end of a voltage source to the low-potential end.
o In reality, it is negatively-charged particles (electrons) that move in a circuit, from low potential to high potential.
· Current flows only in conductive materials.
o Metallic conduction relies on uniform movement of free electrons in metallic bonds.
o Electrolytic conduction relies on the ion concentration of a solution.
o Insulators are materials that do not conduct a current.
· Kirchhoff’s laws express conservation of charge and energy.
o Kirchhoff’s junction rule states that the sum of currents directed into a point within a circuit equals the sum of the currents directed away from that point.
o Kirchhoff’s loop rule states that in a closed loop, the sum of voltage sources is always equal to the sum of voltage drops.
Resistance
· Resistance is opposition to the movement of electrons through a material.
· Resistors are conductive materials with a moderate amount of resistance that slow down electrons without stopping them.
· Resistance is calculated using the resistivity, length, and cross-sectional area of the material in question.
· Ohm’s law states that for a given resistance, the magnitude of the current through a resistor is proportional to the voltage drop across the resistor.
· Resistors in circuits can be combined to calculate the equivalent resistance of a full or partial circuit.
o Resistors in series are additive and sum together to create the total resistance of a circuit.
o Resistors in parallel cause a decrease in equivalent resistance of a circuit.
· Across each resistor in a circuit, a certain amount of power is dissipated, which is dependent on the current through the resistor and the voltage drop across the resistor.
Capacitance and Capacitors
· Capacitors have the ability to store and discharge electrical potential energy.
· Capacitance in parallel plate capacitors is determined by the area of the plates and the distance between the plates.
· Capacitors in series cause a decrease in the equivalent capacitance of a circuit.
· Capacitors in parallel sum together to create a larger equivalent capacitance.
· Dielectric materials are insulators placed between the plates of a capacitor that increase the capacitance of the capacitor by a factor equal to the material’s dielectric constant, κ.
Meters
· Ammeters are inserted in series in a circuit to measure current; they have negligible resistance.
· Voltmeters are inserted in parallel in a circuit to measure a voltage drop; they have very large resistances.
· Ohmmeters are inserted around a resistive element to measure resistance; they are self-powered and have negligible resistance.
Answers to Concept Checks
· 6.1
1. Current is the movement of positive charge through a conductive material over time and is given in ampères . Voltage is a potential difference between two points and is given in volts . Electromotive force (emf) refers to the potential difference of the voltage source for a circuit, usually a battery, and is given in volts. Conductivity is the reciprocal of resistance and is a measure of permissiveness to current flow; it is measured in siemens (S).
2. The sodium chloride solution likely has a higher conductivity because it is a salt and will increase the ion content of water. Glucose does not dissociate, and therefore it has a near-zero impact on conductivity.
3. True. This is a restatement of Kirchhoff’s junction rule.
4. False. While the voltage sources and voltage drops are equal in any closed loop, this is not necessarily true for the entire circuit. For example, a 9 V battery that powers 10 light bulbs in parallel has a 9 V voltage source and a 9 V drop across each light bulb—a total of 90 V of drop across all of the light bulbs combined.
· 6.2
1. Adding a resistor in series increases the total resistance of a circuit; removing one in series decreases the total resistance in the circuit. These relationships are reversed in parallel: adding a resistor decreases resistance while removing one increases it.
2. Resistivity, length, cross-sectional area, and temperature all contribute to the resistance of a resistor.
3. Power is related to current, voltage, and resistance through the equations
4. True. The internal resistance will lower the available voltage for the circuit. Lowering the available voltage will also lower current for any given resistance.
· 6.3
1. The capacitor discharges, providing a current in the opposite direction of the initial current.
2. A dielectric material will always increase capacitance. If the capacitor is isolated, its voltage will decrease when a dielectric material is introduced; if it is in a circuit, its voltage is constant because it is dictated by the voltage source. If a capacitor is isolated, the stored charge will remain constant because there is no additional source of charge; if it is in a circuit, the stored charge will increase.
3. Adding a capacitor in series decreases the total capacitance of a circuit; removing one in series increases the total capacitance in the circuit. These relationships are reversed in parallel: adding a capacitor increases resistance while removing one increases it.
4. Surface area, distance, and dielectric constant all contribute to the capacitance of a capacitor.
· 6.4
1.
Meter Type |
Measures… |
Placement |
Ideal Resistance |
Ammeter |
Current |
In series with point of interest |
0 |
Voltmeter |
Potential difference (voltage) |
Parallel with circuit element of interest |
∞ |
Ohmmeter |
Resistance |
Two points in series with circuit element of interest |
0 |
2. False. Voltmeters and ammeters are designed to have minimum impact on a circuit, thus they can be used together.
Equations to Remember
(6.1) Current:
(6.2) Kirchhoff’s junction rule: I_{into junction} = I_{leaving junction}
(6.3) Kirchhoff’s loop rule: V_{source} = V_{drop}
(6.4) Definition of resistance:
(6.5) Ohm’s law: V = IR
(6.6) Voltage and cell emf: V = E_{cell} – ir_{int}
(6.7) Definition of power:
(6.8) Electric power:
(6.9) Voltage drop across circuit elements (series): V_{s} = V_{1} + V_{2} + V_{3} + ⋯ + Vn
(6.10) Equivalent resistance (series): R_{s} = R_{1} + R_{2} + R_{3} + ⋯ + Rn
(6.11) Voltage drop across circuit elements (parallel): V_{p} = V_{1} = V_{2} = V_{3} = ⋯ = Vn
(6.12) Equivalent resistance (parallel):
(6.13) Definition of capacitance:
(6.14) Capacitance based on parallel plate geometry:
(6.15) Electric field in a capacitor:
(6.16) Potential energy of a capacitor:
(6.17) Capacitance with a dielectric material: C′ = κC
(6.18) Equivalent capacitance (series):
(6.19) Equivalent capacitance (parallel): C_{p} = C_{1} + C_{2} + C_{3} + ⋯ + C_{n}
Shared Concepts
· Biology Chapter 6
o The Respiratory System
· Biology Chapter 7
o The Cardiovascular System
· General Chemistry Chapter 12
o Electrochemistry
· Physics and Math Chapter 2
o Work and Energy
· Physics and Math Chapter 4
o Fluids
· Physics and Math Chapter 5
o Electrostatics and Magnetism