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Current, Resistance, Direct-current Circuits

Current, Resistance, Direct-current Circuits. 1. Electric current The current is the rate at which charge flows through this surface

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Current, Resistance, Direct-current Circuits

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  1. Current, Resistance, Direct-current Circuits

  2. 1. Electric current • The currentis the rate at which charge flows through this surface • Suppose ΔQ is the amount of charge that flows through an area A in a time interval Δt and that the direction of flow is perpendicular to the area. Then the current I is equal to the amount of charge divided by the time • I= Δ Q/Δt • SI C/s=A (Ampere)

  3. 2. Current and voltage measuraments in circuits

  4. 3. Resistance and Ohm’s Law • Resistance is the ratio of the voltage across the conductor to the current: • R = ΔV/ I • SI units :1V/A=1Ω (Ohms) • Ohm’s Law: the resistance remains constant over a wide range of applied voltage or currents: • ΔV =IR

  5. A resistor : is a conductor that provides a specified resistance in an electric circuit • The resistance is proportional to the conductor’s length l and inversely proportional to its cross-sectional area A: • R =ρ l/A • ρ –resistivity of the material (ct. of proportionality) SI unit: Ω m

  6. 4. Electrical energy and power • The rate at which the system loses potential energy as the charge passes through resistor is equal to the rate at which the system gains internal energy in the resistor. • (ΔQ/ Δt) ΔV =I ΔV • Power P - the rate at which energy is delivered to the resistor: • P =I ΔV • P =I2R = ΔV2/R • SI: kilowatt-hour • 1kWh =(103W)(3600s) =3.6x106J

  7. Question: Some homes have light dimmers that are operated by rotating a knob. What is being changed in the electric circuit when the know is rotated? R I

  8. Direct-Current Circuits 5. Sources of EMF (electromotive force)- a “ charge pump” that forces electrons to move in a direction opposite the electrostatic field inside the source • The emf εof a source is the work done per unit charge (SI unit: V) • ΔV =ε –Ir • ε= the terminal voltage when the current is zero (open circuit voltage)

  9. ΔV= IR (R- the external resistance) • ε =IR +Ir; • I = ε / (R+r) • I ε =I2R + I2r if r<<R • (the power delivered • by the battery is • transferred to the load • resistance)

  10. 6. Resistors in series

  11. ΔV = IR1 +IR2 =I(R1+R2) • ΔV = I Req • IReq = I(R1+R2) • Req =R1+R2 • Req =R1+R2+…. • The equivalent resistanceof a series combination of resistors is the algebraic sum of the individual resistances and is always greater than any individual resistance

  12. 7. Resistors in parallel

  13. The potential differences across resistors are the same because each is connected directly across the battery terminals • Because charge is conserved, the current I that enters point a must equal the total current I1+I2 leaving that point • I =I1+I2 • I1=ΔV /R1 ; • I2=ΔV /R2 ; • I =ΔV /Req ; • 1/Req = 1/R1 +1/R2 • The inverse of the equivalent resistance of two or more resistors connected in parallel is the sum of the inverses of the individual resistances and is always less than the smallest resistance in the group

  14. 8. Kirchhoff’s Rules: • 1. The sum of the currents entering any jonction must equal the sum of the currents leaving that junction (junction rule)

  15. 2. The sum of the potential differences across all the elements around any closed circuit loop must be zero (loop rule)

  16. a) if the resistor is traversed in the direction of the current, the charge in electric potential across the resistor is –IR b) If a resistor is traversed in the direction opposite the current, the charge in electric potential across resistors is +IR

  17. c) if a source of emf is traversed in the direction of the emf (from – to +) the charge in electric potential is +Σ d) if a source of emf is traversed in the direction opposite the emf, the charge in electric potential is -Σ

  18. Problem solving strategy: • 1. Assign labels and symbols to all the known and unknown quantities • 2. Assign directions to the currents in each part of the circuit • 3. Apply the junction rule to any junction in the circuit • 4. Apply Kirchhoff’s loop rule, to as many loops in the circuit as are needed to solve for the unknowns • 5. solve the equations • 6. Check your answers by substituting them into the original equations

  19. Problem: Find the currents in the circuit shown in fig. by using Kirchhoff’s rules

  20. 9. RC circuits

  21. We consider that the capacitor is initially uncharged with the switch open. After the switch is closed, the battery begins to charge the plates of capacitor and the charge passes through resistor • A the capacitor is being charged, the circuit carries a changing current. The process continues until the capacitor is charged to a maximum value • Q=Cε • ε -maximum voltage across the capacitor • Once the capacitor is fully charged, the current in circuit is zero

  22. q=Q(1-e –t/RC) • RC=σ –time constant • The time constant represents the time required for the charge to increase from zero to 63.2% its maximum equilibrium value

  23. Before the switch is closed the potential difference across the charged capacitor is Q/C • Once the switch is closed, the charge begins to flow through the resistor, until the capacitor is fully charged • q=Qe-t/RC • The charge decreasses exponentially with time • ΔV=εe-t/RC

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