Electricity and Magnetism - Past Notes PDF

Electricity and Magnetism Topic 1.5: Current Resistance Ohm’s Law Electromotive Force Circuits Based from Sears and Zemansky’s University Physics With Modern Physics (13th ed) Current Any motion of charge from one region to another For a conductor without...

Electricity and Magnetism Topic 1.5: Current Resistance Ohm’s Law Electromotive Force Circuits Based from Sears and Zemansky’s University Physics With Modern Physics (13th ed) Current Any motion of charge from one region to another For a conductor without internal E field, motion is random, and no electron escapes the conductor; hence, there no NET flow = no current For a conductor with internal E field, the free electrons are subjected to a steady force and accelerates to the direction of the force, but they can also collide with massive ions making these free electrons to change direction. This slow net motion (as a group) is called drift motion (~10-4 m/s). Direction of Current Note: Current is not a vector. In a current-carrying wire, the current is always along the length of the wire. No single vector can describe motion along a curved path (when wire is curved); hence, current is not a vector Current Current Density (J) – current (I) per unit cross-sectional area (A) 𝐼 Where vd is the drift velocity 𝐽 = = 𝑛 𝑞 𝑣𝑑 q is the magnitude of the charge 𝐴 n is the concentration of particles SI Unit: 1 C/s∙m2 = 1 A/m2 General Expression for Current - Net charge flowing through the area per unit time 𝑑𝑄 𝐼= = 𝑛 𝑞 𝑣𝑑 𝐴 𝑑𝑡 SI Unit: Coulomb/second [C/s] = Ampere [A] Current Copper has 8.5x1028 free electrons per 𝑣𝑑 = 1.079𝑥10−4 𝑚/𝑠 cubic meter. A 71.0-cm length of 12- We can use the eq’n for uniform gauge copper wire that is 2.05 mm in motion from classical mech v= dt diameter carries 4.85 A of current. Δ𝑡 = 𝐿𝑒𝑛𝑔𝑡ℎ/𝑣𝑑 (a) How much time does it take for an electron to travel the length of wire? 0.710𝑚 Δ𝑡 = = 6.58𝑥103 𝑠 1.079𝑚/𝑠 𝐼 4.85 𝐴 𝐽= = = 1.469𝑥106 𝐴/𝑚2 𝐴 2.05𝑥10−3 𝑚 2 E1. Repeat part (a) for 6- 𝜋 gauge copper wire 2 (diameter 4.12 mm) of the 𝐴 𝐽 1.469𝑥106 𝑚2 same length that carries the 𝐽 = 𝑛 𝑞 𝑣𝑑 → 𝑣𝑑 = = same current. Ans: 2.66x104 s 𝑛𝑞 8.5𝑥1028 1.602𝑥1019 𝐶 Ohm’s Law States that current density is dependent (directly proportional to) on electric field and their ratio is constant, but in general this dependence is complex. Named after German physicist Georg Simon Ohm Note: Like the Ideal Gas Equation and Hooke’s Law, is an idealized model that describes the behavior of some materials quite well but is not a general description of all matter. A material that obeys Ohm’s Law is called ohmic or a linear conductor. Resistivity (ρ) Ratio of the magnitudes of electric field and current density. Good conductors have small resistivity, while good insulators have high resistivity. Resistivity depends on the kind of material. 𝐸 Where E is electric field and J is current density 𝜌= 𝐽 SI Unit: 1 Nm2/C∙A = 1 Ω∙m (read as “Ohm meter” Conductivity – reciprocal of resistivity (SI Unit: 1/ Ω∙m ) Resistivity and Temperature For conductors, resistivity increases with temperature. For small temperature changes, resistivity as a function of temperature (T) is: 𝜌(𝑇) = 𝜌0 [1 + 𝛼(𝑇 − 𝑇0 ] Where α is temperature coefficient of resistivity Thermistor thermometer uses this principle to have a sensitive measure of temperature. Resistivity (ρ) Resistivities at Room Temperature Resistivity and Temperature E2. A constant E field is maintained inside a piece of semiconductor while lowering the its temperature. What happens to the current density in the semiconductor? A. It increases B. It decreases. C. It remains the same. Resistance Relationship between resistance and resistivity 𝜌𝐿 𝑅= If ρ is constant, then so is R. 𝐴 Where L and A are the length and cross- sectional area of the conductor, respectively Relationship among voltage, current, and resistance 𝑉 𝑅= SI Unit: 1 Ohm [Ω] = 1 [V/A] 𝐼 This is often called Ohm’s Law– the direct proportionality of V to I or J to E for some materials. Resistance Resistance as a function of temperature 𝑅 𝑇 = 𝑅0 [1 + 𝛼(𝑇 − 𝑇0 ) This is analogous to the previous equation of resistivity as a function of T (again only for temperature changes that are not too great). Resistor – a circuit device made to have a specific value of resistance between its ends. Resistance A wire 6.60 m long with diameter of 2.05 mm has a resistance of 0.0290 Ohm. What material is the wire most likely made of? To solve this, we find the resistivity and compare the value to those in the table. 𝐿 𝑅𝐴 E3. Based on the computed value, 𝑅=𝜌 →𝜌= identify the material that the wire is 𝐴 𝐿 0.0290Ω 1.025𝑥103 𝑚 2 𝜋 most likely made of. 𝜌= 6.60 𝑚 𝜌 = 1.45𝑥10−8 Ω𝑚 Ohm’s Law Current-voltage relationship for two devices (ohmic and nonohmic resistor) Current, Resistance, Voltage E4. Suppose you increase the voltage across the copper wire. The increased voltage causes more current to flow, which make the temperature of the wire increase. If you double the voltage across the wire, the current in the wire increases. By what factor does it increase? a) 2 b) greater than 2 c) less than 2 Electromotive Force Steady current is established in a conductor when the conductor is part of a complete circuit (closed loop). Charges moves around a closed loop and returns to its starting point which means the potential energy at the beginning is the same at the end. So there must be some part in the circuit where the potential energy increases. This analogous to a water fountain that recycles its water. Water cascades down (decreases potential energy) but is lifted back to the top by a pump (increases potential energy). Electromotive Force (ξ) The device that acts like a pump in the circuit is called an EMF source. Examples: battery, generators, solar cells, thermocouples, and fuel cells NOTE: Electromotive force is NOT a force but an energy per unit charge quantity (same unit as in potential difference). “Force” is used only because it acts like a “force” that influences current to flow “uphill” from lower to higher potential (negative to positive terminal of a battery). Electromotive Force (ξ) For an ideal EMF source 𝑉𝑎𝑏 = 𝜉 = 𝐼𝑅 For a real EMF source 𝑉𝑎𝑏 = 𝜉 − 𝐼𝑟 Terminal Voltage (Vab) < ξ because of internal resistance (r)—resistance in the EMF source. “Ir” is the voltage drop. Open circuit Closed circuit Electromotive Force (ξ) An ideal voltmeter V is connected to a 2.0-Ω resistor and a battery with emf 5.0 V and internal resistance 0.5 Ω. What is the current in the 2.0- Ω resistor? What is the terminal voltage of the battery? What is the reading on the voltmeter? a) Since an ideal voltmeter has infinite resistance, there would be no current through the external resistor. b) Since there is no current, there would be no voltage drop across the battery; hence the terminal voltage is equal to emf = 5.0 V c) Voltmeter reads 5.0 V because there is no voltage drop across both resistors. Symbols for Circuit Diagrams Energy and Power in Circuits Rate at which energy is delivered to or extracted from a circuit element 𝑃 = 𝑉𝑎𝑏 𝐼 SI Unit: Watt [W] = J/s = V∙A = A2Ω = V2/ Ω Power delivered to a resistor 2 𝑉𝑎𝑏 𝑃 = 𝑉𝑎𝑏 𝐼 = 𝐼2 𝑅 = 𝑅 Potential changes around a circuit. Energy and Power in Circuits A resistor with resistance R is connected to a battery that has emf 12.0 V and internal resistance r=0.40 Ω. For what two values of R will the power dissipated in the resistor be 80.0 W? 2𝑅 𝐼= 𝜉 and 𝑃 = 𝐼 𝑅2 𝜉 𝑅+𝑟 𝑃 𝑅2 + 2𝑅𝑟 − + 𝑟2 = 0 𝑃 𝜉 2 𝑃= 𝑅 Quadratic Formula 𝑅+𝑟 𝜉2𝑅 −𝑏 ± 𝑏 2 − 4𝑎𝑐 𝑃 = 𝑅2 +2𝑅𝑟+𝑟 2 𝑥= 2𝑎 𝜉2𝑅 = 𝑃(𝑅2 + 2𝑅𝑟 + 𝑟 2) 80 ± 802 − 4(80)(12.8) 𝑅= 2(80) 𝑃 𝑅2 + 2𝑅𝑟 + 𝑟 2 − 𝜉 2 𝑅 = 0 𝑅 = 0.2 Ω and 0.8 Ω Energy and Power in Circuits A cylindrical copper cable 1500 m long is connected across a 220-V potential difference. A) What should be its diameter so that it produces heat at 75.0 W? 𝜌𝐿 B) What is the electric field inside these 𝑃= 𝑉 2 /𝑅 𝑅= 𝐴 conditions? 𝑉2 2 𝐴 𝑉 220𝑉 𝑃= =𝑉 = (220𝑉) 𝐸= = = 0.147 𝑉/𝑚 𝑅 𝜌𝐿 𝐿 1500𝑚 2 𝜋𝑟 2 75𝑊 = 220𝑉 1.72𝑥10−8 Ω ⋅ 𝑚𝑥1500𝑚 E5. Compute diameter. Ans: 0.184 mm