Lecture 09: Ohm's Law PDF

Ohm’s Law Angel Ramoso Lecture 09 Angel Ramoso 1/ 29 Electrodynamics Lecture 09 Angel Ramoso 2/ 29 Electrodynamics ˆ Electrostatics Lecture 09 Angel Ramoso 2/ 29 Electrodynamics ˆ Electrostatics ⇒ Deals with the interaction of electric charges at r...

Ohm’s Law Angel Ramoso Lecture 09 Angel Ramoso 1/ 29 Electrodynamics Lecture 09 Angel Ramoso 2/ 29 Electrodynamics ˆ Electrostatics Lecture 09 Angel Ramoso 2/ 29 Electrodynamics ˆ Electrostatics ⇒ Deals with the interaction of electric charges at rest. Lecture 09 Angel Ramoso 2/ 29 Electrodynamics ˆ Electrostatics ⇒ Deals with the interaction of electric charges at rest. ˆ Electrodynamics Lecture 09 Angel Ramoso 2/ 29 Electrodynamics ˆ Electrostatics ⇒ Deals with the interaction of electric charges at rest. ˆ Electrodynamics ⇒ Deals with electric charges in motion. Lecture 09 Angel Ramoso 2/ 29 Electrodynamics ˆ Electrostatics ⇒ Deals with the interaction of electric charges at rest. ˆ Electrodynamics ⇒ Deals with electric charges in motion. ⇒ Deals with electricity Lecture 09 Angel Ramoso 2/ 29 Electricity Lecture 09 Angel Ramoso 3/ 29 Electricity ˆ Flow of charges Lecture 09 Angel Ramoso 3/ 29 Electricity ˆ Flow of charges ˆ Circuit → closed path for charges Lecture 09 Angel Ramoso 3/ 29 Electricity ˆ Flow of charges ˆ Circuit → closed path for charges ˆ Current → Positive charge → move from + side to the − side (lower potential) Lecture 09 Angel Ramoso 3/ 29 Electricity ˆ Flow of charges ˆ Circuit → closed path for charges ˆ Current → Positive charge → move from + side to the − side (lower potential) Lecture 09 Angel Ramoso 3/ 29 Electricity ˆ Flow of charges ˆ Circuit → closed path for charges ˆ Current → Positive charge → move from + side to the − side (lower potential) Lecture 09 Angel Ramoso 3/ 29 Circuits Lecture 09 Angel Ramoso 4/ 29 Circuits ˆ Most electronic devices, if not all, are composed of circuits. Lecture 09 Angel Ramoso 4/ 29 Circuits ˆ Most electronic devices, if not all, are composed of circuits. ˆ One could say circuits are the building blocks of the modern world. Lecture 09 Angel Ramoso 4/ 29 Circuits’ Basic Parts Lecture 09 Angel Ramoso 5/ 29 Circuits’ Basic Parts ˆ Power Supply ˆ Wire (conductor for the charges to move → electricity) ˆ Capacitor ˆ Resistor ˆ Switch ˆ Load ˆ Others: Transistors, Diodes, Inductors, Transformers, etc Lecture 09 Angel Ramoso 5/ 29 Current Lecture 09 Angel Ramoso 6/ 29 Current ˆ (Electric) Circuit Lecture 09 Angel Ramoso 6/ 29 Current ˆ (Electric) Circuit ⇒ Closed path of conducting loop where electric charges move. Lecture 09 Angel Ramoso 6/ 29 Current ˆ (Electric) Circuit ⇒ Closed path of conducting loop where electric charges move. ˆ (Electric) Current Lecture 09 Angel Ramoso 6/ 29 Current ˆ (Electric) Circuit ⇒ Closed path of conducting loop where electric charges move. ˆ (Electric) Current ⇒ Consists of charges in motion from one region to another. Lecture 09 Angel Ramoso 6/ 29 Current ˆ (Electric) Circuit ⇒ Closed path of conducting loop where electric charges move. ˆ (Electric) Current ⇒ Consists of charges in motion from one region to another. Lecture 09 Angel Ramoso 6/ 29 Current ˆ Any motion of charge from one region to another. Lecture 09 Angel Ramoso 7/ 29 Current ˆ Any motion of charge from one region to another. ˆ Recall, in electrostatic situations: Lecture 09 Angel Ramoso 7/ 29 Current ˆ Any motion of charge from one region to another. ˆ Recall, in electrostatic situations: ⇒ Electric field is zero everywhere within the conductor. Lecture 09 Angel Ramoso 7/ 29 Current ˆ Any motion of charge from one region to another. ˆ Recall, in electrostatic situations: ⇒ Electric field is zero everywhere within the conductor. −→ No current. Lecture 09 Angel Ramoso 7/ 29 Current ˆ Any motion of charge from one region to another. ˆ Recall, in electrostatic situations: ⇒ Electric field is zero everywhere within the conductor. −→ No current. ⇒ Motion of the electrons is random. Lecture 09 Angel Ramoso 7/ 29 Current ˆ Any motion of charge from one region to another. ˆ Recall, in electrostatic situations: ⇒ Electric field is zero everywhere within the conductor. −→ No current. ⇒ Motion of the electrons is random. −→ No net flow of charge in any direction and hence no current. Lecture 09 Angel Ramoso 7/ 29 Current Lecture 09 Angel Ramoso 8/ 29 Current ⇀ ˆ Recall: In electrostatics, E = 0 inside the conductor Lecture 09 Angel Ramoso 8/ 29 Current ⇀ ˆ Recall: In electrostatics, E = 0 inside the conductor ⇒ No net flow of charge. Lecture 09 Angel Ramoso 8/ 29 Current ⇀ ˆ Recall: In electrostatics, E = 0 inside the conductor ⇒ No net flow of charge. Lecture 09 Angel Ramoso 8/ 29 Current ⇀ ˆ Recall: In electrostatics, E = 0 inside the conductor ⇒ No net flow of charge. Lecture 09 Angel Ramoso 8/ 29 Current ⇀ ˆ Recall: In electrostatics, E = 0 inside the conductor ⇒ No net flow of charge. Paths — motion of charges in the tubes. Lecture 09 Angel Ramoso 8/ 29 Current ⇀ ˆ Recall: In electrostatics, E = 0 inside the conductor ⇒ No net flow of charge. Paths — motion of charges in the tubes. Blue — w/o electric field (random motion). Lecture 09 Angel Ramoso 8/ 29 Current What happens if a constant, steady electric field is established inside a conductor? Paths — motion of charges in the tubes. Blue — w/o electric field (random motion). Lecture 09 Angel Ramoso 9/ 29 Current What happens if a constant, steady electric field is established inside a conductor? Paths — motion of charges in the tubes. Blue — w/o electric field (random motion). Lecture 09 Angel Ramoso 9/ 29 Current What happens if a constant, steady electric field is established inside a conductor? Paths — motion of charges in the tubes. Blue — w/o electric field (random motion). Lecture 09 Angel Ramoso 9/ 29 Current What happens if a constant, steady electric field is established inside a conductor? Paths — motion of charges in the tubes. Blue — w/o electric field (random motion). Lecture 09 Angel Ramoso 9/ 29 Current What happens if a constant, steady electric field is established inside a conductor? Paths — motion of charges in the tubes. Blue — w/o electric field (random motion). Pink — w/ electric field (has net displacement). Lecture 09 Angel Ramoso 9/ 29 Current What happens if a constant, steady electric field is established inside a conductor? Paths — motion of charges in the tubes. Blue — w/o electric field (random motion). Pink — w/ electric field (has net displacement). *The drift is exaggerated in this figure. Lecture 09 Angel Ramoso 9/ 29 Current What happens if a constant, steady electric field is established inside a conductor? Paths — motion of charges in the tubes. Blue — w/o electric field (random motion). Pink — w/ electric field (has net displacement). ⇀v d is called the drift velocity of *The drift is exaggerated in this figure. the particle. Lecture 09 Angel Ramoso 9/ 29 Current Direction of conventional current Lecture 09 Angel Ramoso 10/ 29 Current Direction of conventional current Convention in the direction — Current is the flow of positive charges, positive charges move in the direction of electric field. Lecture 09 Angel Ramoso 10/ 29 Current Direction of conventional current Convention in the direction — Current is the flow of positive charges, positive charges move in the direction of electric field. Lecture 09 Angel Ramoso 10/ 29 Current Direction of conventional current Convention in the direction — Current is the flow of positive charges, positive charges move in the direction of electric field. In this figure, the current I points to the right. Lecture 09 Angel Ramoso 10/ 29 Current Current in Metals Lecture 09 Angel Ramoso 11/ 29 Current Current in Metals In metals (where charges are negative), drift velocity is opposite to the electric field. Lecture 09 Angel Ramoso 11/ 29 Current Current in Metals In metals (where charges are negative), drift velocity is opposite to the electric field. Lecture 09 Angel Ramoso 11/ 29 Current Current in Metals In metals (where charges are negative), drift velocity is opposite to the electric field. However, direction of current still follows the convention. Lecture 09 Angel Ramoso 11/ 29 Current dq I = ∼ (1) dt Lecture 09 Angel Ramoso 12/ 29 Current dq I = ∼ (1) dt Lecture 09 Angel Ramoso 12/ 29 Current dq I = ∼ (1) dt ˆ Current is defined as the net charge flowing through an area per unit time Lecture 09 Angel Ramoso 12/ 29 Current dq I = ∼ (1) dt ˆ Current is defined as the net charge flowing through an area per unit time ˆ SI unit: Ampere ( [A]) Lecture 09 Angel Ramoso 12/ 29 Current dq I = ∼ (1) dt ˆ Current is defined as the net charge flowing through an area per unit time ˆ SI unit: Ampere ( [A])   C ⇒ 1 [A] = 1 s Lecture 09 Angel Ramoso 12/ 29 Resistivity and Conductivity Lecture 09 Angel Ramoso 13/ 29 Resistivity and Conductivity ⇀ As E increases, the charges move faster, and so the current density ⇀ ⇀ J also increases where J = I /A. The ratio ρ is a constant. Lecture 09 Angel Ramoso 13/ 29 Resistivity and Conductivity ⇀ As E increases, the charges move faster, and so the current density ⇀ ⇀ J also increases where J = I /A. The ratio ρ is a constant. ⇀ ⇀ Resistivity (ρ) — ratio between E and J of a material Lecture 09 Angel Ramoso 13/ 29 Resistivity and Conductivity ⇀ As E increases, the charges move faster, and so the current density ⇀ ⇀ J also increases where J = I /A. The ratio ρ is a constant. ⇀ ⇀ Resistivity (ρ) — ratio between E and J of a material ⇀ ⇀ E =ρ J ∼ (2) Lecture 09 Angel Ramoso 13/ 29 Resistivity and Conductivity ⇀ As E increases, the charges move faster, and so the current density ⇀ ⇀ J also increases where J = I /A. The ratio ρ is a constant. ⇀ ⇀ Resistivity (ρ) — ratio between E and J of a material ⇀ ⇀ E =ρ J ∼ (2) Conductivity (σ) — reciprocal of the resistivity Lecture 09 Angel Ramoso 13/ 29 Resistivity and Conductivity ⇀ As E increases, the charges move faster, and so the current density ⇀ ⇀ J also increases where J = I /A. The ratio ρ is a constant. ⇀ ⇀ Resistivity (ρ) — ratio between E and J of a material ⇀ ⇀ E =ρ J ∼ (2) Conductivity (σ) — reciprocal of the resistivity ⇀ ⇀ J =σE ∼ (3) 1 σ= ∼ (4) ρ Lecture 09 Angel Ramoso 13/ 29 Resistivity and Conductivity Resistivity (ρ) E ρ= ∼ (5) J Conductivity (σ) 1 J σ= = ∼ (6) ρ E Lecture 09 Angel Ramoso 14/ 29 Resistivity and Conductivity Resistivity (ρ)   V·m E SI unit: ρ= ∼ (5) A J Conductivity (σ) 1 J σ= = ∼ (6) ρ E Lecture 09 Angel Ramoso 14/ 29 Resistivity and Conductivity Resistivity (ρ)   V·m E SI unit: ρ= ∼ (5) A J Conductivity (σ)   A 1 J SI unit: σ= = ∼ (6) V·m ρ E Lecture 09 Angel Ramoso 14/ 29 Resistivity and Conductivity Resistivity (ρ)   V·m E SI unit: ρ= ∼ (5) A J Conductivity (σ)   A 1 J SI unit: σ= = ∼ (6) V·m ρ E   V Note: 1 = 1 [Ω] A Lecture 09 Angel Ramoso 14/ 29 Resistivity and Conductivity Resistivity (ρ) E SI unit: [Ω · m] ρ= ∼ (5) J Conductivity (σ)   A 1 J SI unit: σ= = ∼ (6) V·m ρ E   V Note: 1 = 1 [Ω] A Lecture 09 Angel Ramoso 14/ 29 Resistivity and Conductivity Resistivity (ρ) E SI unit: [Ω · m] ρ= ∼ (5) J Conductivity (σ) h
−1 i 1 J SI unit: Ω·m σ= = ∼ (6) ρ E   V Note: 1 = 1 [Ω] A Lecture 09 Angel Ramoso 14/ 29 Resistivity and Conductivity ˆ For perfect conductors, ρ → 0 and σ → ∞ ˆ For perfect insulators, ρ → ∞ and σ → 0 Lecture 09 Angel Ramoso 15/ 29 Resistance Lecture 09 Angel Ramoso 16/ 29 Resistance Different objects but geometrically similar connected with the same potential difference can yield different currents. Lecture 09 Angel Ramoso 16/ 29 Resistance Different objects but geometrically similar connected with the same potential difference can yield different currents. ↓ Lecture 09 Angel Ramoso 16/ 29 Resistance Different objects but geometrically similar connected with the same potential difference can yield different currents. ↓ resistance Lecture 09 Angel Ramoso 16/ 29 Resistance Different objects but geometrically similar connected with the same potential difference can yield different currents. ↓ resistance V R= or V = IR ∼ (7) I with SI units: [Ω]. Lecture 09 Angel Ramoso 16/ 29 Resistance Different objects but geometrically similar connected with the same potential difference can yield different currents. ↓ resistance V R= or V = IR ∼ (7) I with SI units: [Ω]. A conductor with this function is called a resis- tor. Lecture 09 Angel Ramoso 16/ 29 Resistance Different objects but geometrically similar connected with the same potential difference can yield different currents. ↓ resistance V R= or V = IR ∼ (7) I with SI units: [Ω]. A conductor with this function is called a resis- tor. Symbol: Lecture 09 Angel Ramoso 16/ 29 Resistance Different objects but geometrically similar connected with the same potential difference can yield different currents. ↓ resistance V R= or V = IR ∼ (7) I with SI units: [Ω]. A conductor with this function is called a resis- tor. Symbol: Note: Resistance is a property of an object. Resistivity is a property of a material. Lecture 09 Angel Ramoso 16/ 29 Ohm’s Law Lecture 09 Angel Ramoso 17/ 29 Ohm’s Law E V ρ= and R= ∼ (8) J I When the resistivity is constant, this is called Ohm’s law. Lecture 09 Angel Ramoso 17/ 29 Ohm’s Law E V ρ= and R= ∼ (8) J I When the resistivity is constant, this is called Ohm’s law. ˆ Ohmic conductors or linear conductors Lecture 09 Angel Ramoso 17/ 29 Ohm’s Law E V ρ= and R= ∼ (8) J I When the resistivity is constant, this is called Ohm’s law. ˆ Ohmic conductors or linear conductors ⇒ Materials that obeys Ohm’s law Lecture 09 Angel Ramoso 17/ 29 Ohm’s Law E V ρ= and R= ∼ (8) J I When the resistivity is constant, this is called Ohm’s law. ˆ Ohmic conductors or linear conductors ⇒ Materials that obeys Ohm’s law ˆ Nonohmic or nonlinear material Lecture 09 Angel Ramoso 17/ 29 Ohm’s Law E V ρ= and R= ∼ (8) J I When the resistivity is constant, this is called Ohm’s law. ˆ Ohmic conductors or linear conductors ⇒ Materials that obeys Ohm’s law ˆ Nonohmic or nonlinear material ⇒ Material that does not obey Ohm’s law Lecture 09 Angel Ramoso 17/ 29 Ohm’s Law Ohm’s law is an assertion that the current through a device is always directly proportional to the potential difference applied to the device. Lecture 09 Angel Ramoso 18/ 29 Ohm’s Law Ohm’s law is an assertion that the current through a device is always directly proportional to the potential difference applied to the device. For real objects, temperature affects resistivity too. Lecture 09 Angel Ramoso 18/ 29 Ohm’s Law Ohm’s law is an assertion that the current through a device is always directly proportional to the potential difference applied to the device. For real objects, temperature affects resistivity too. ˆ Resistivity depends on Lecture 09 Angel Ramoso 18/ 29 Ohm’s Law Ohm’s law is an assertion that the current through a device is always directly proportional to the potential difference applied to the device. For real objects, temperature affects resistivity too. ˆ Resistivity depends on ⇒ Type of the material Lecture 09 Angel Ramoso 18/ 29 Ohm’s Law Ohm’s law is an assertion that the current through a device is always directly proportional to the potential difference applied to the device. For real objects, temperature affects resistivity too. ˆ Resistivity depends on ⇒ Type of the material ⇒ Temperature of the material Lecture 09 Angel Ramoso 18/ 29 Ohm’s Law Ohm’s law is an assertion that the current through a device is always directly proportional to the potential difference applied to the device. For real objects, temperature affects resistivity too. ˆ Resistivity depends on ⇒ Type of the material ⇒ Temperature of the material ˆ Resistivity does NOT depend on Lecture 09 Angel Ramoso 18/ 29 Ohm’s Law Ohm’s law is an assertion that the current through a device is always directly proportional to the potential difference applied to the device. For real objects, temperature affects resistivity too. ˆ Resistivity depends on ⇒ Type of the material ⇒ Temperature of the material ˆ Resistivity does NOT depend on ⇒ Size and shape of the material Lecture 09 Angel Ramoso 18/ 29 Ohm’s Law For conductors, Lecture 09 Angel Ramoso 19/ 29 Ohm’s Law For semiconductors, Lecture 09 Angel Ramoso 20/ 29 Ohm’s Law For superconductors, Lecture 09 Angel Ramoso 21/ 29 Resistivity Exercise Lecture 09 Angel Ramoso 22/ 29 Resistivity Exercise A copper wire (with resistivity 1.69 × 10−8 [Ω · m]) has a cross- sectional area of 1.31 mm2. It carries a current of 2.00 [A]. Find   the electric field magnitude in the wire. Lecture 09 Angel Ramoso 22/ 29 Resistivity Solution Lecture 09 Angel Ramoso 23/ 29 Resistivity Solution Electric Field magnitude Lecture 09 Angel Ramoso 23/ 29 Resistivity Solution Electric Field magnitude E ρ= ∼ (9) J Lecture 09 Angel Ramoso 23/ 29 Resistivity Solution Electric Field magnitude E ρ= ∼ (9) J E = ρJ ∼ (10) Lecture 09 Angel Ramoso 23/ 29 Resistivity Solution Electric Field magnitude E ρ= ∼ (9) J E = ρJ ∼ (10)   I E =ρ ∼ (11) A Lecture 09 Angel Ramoso 23/ 29 Resistivity Solution Electric Field magnitude E ρ= ∼ (9) J E = ρJ ∼ (10)   I E =ρ ∼ (11) A   N E = 0.02580152672 ∼ (12) C Lecture 09 Angel Ramoso 23/ 29 Resistance Lecture 09 Angel Ramoso 24/ 29 Resistance For a conductor of resistivity ρ, Lecture 09 Angel Ramoso 24/ 29 Resistance For a conductor of resistivity ρ, ⇀ ⇀ E =ρ J ∼ (13) Lecture 09 Angel Ramoso 24/ 29 Resistance For a conductor of resistivity ρ, ⇀ ⇀ E =ρ J ∼ (13) Problem: Lecture 09 Angel Ramoso 24/ 29 Resistance For a conductor of resistivity ρ, ⇀ ⇀ E =ρ J ∼ (13) ⇀ ⇀ Problem: E and J are difficult to measure Lecture 09 Angel Ramoso 24/ 29 Resistance For a conductor of resistivity ρ, ⇀ ⇀ E =ρ J ∼ (13) ⇀ ⇀ Problem: E and J are difficult to measure Lecture 09 Angel Ramoso 24/ 29 Resistance For a conductor of resistivity ρ, ⇀ ⇀ E =ρ J ∼ (13) ⇀ ⇀ Problem: E and J are difficult to measure Solution: Lecture 09 Angel Ramoso 24/ 29 Resistance For a conductor of resistivity ρ, ⇀ ⇀ E =ρ J ∼ (13) ⇀ ⇀ Problem: E and J are difficult to measure ⇀ Solution: from E , we can get V , Lecture 09 Angel Ramoso 24/ 29 Resistance For a conductor of resistivity ρ, ⇀ ⇀ E =ρ J ∼ (13) ⇀ ⇀ Problem: E and J are difficult to measure ⇀ ⇀ Solution: from E , we can get V , and from J , we can get I Lecture 09 Angel Ramoso 24/ 29 Resistance For a conductor of resistivity ρ, ⇀ ⇀ E =ρ J ∼ (13) ⇀ ⇀ Problem: E and J are difficult to measure ⇀ ⇀ Solution: from E , we can get V , and from J , we can get I With J and E uniform inside Lecture 09 Angel Ramoso 24/ 29 Resistance For a conductor of resistivity ρ, ⇀ ⇀ E =ρ J ∼ (13) ⇀ ⇀ Problem: E and J are difficult to measure ⇀ ⇀ Solution: from E , we can get V , and from J , we can get I With J and E uniform inside   V I =ρ ∼ (14) L A Lecture 09 Angel Ramoso 24/ 29 Resistance For a conductor of resistivity ρ, ⇀ ⇀ E =ρ J ∼ (13) ⇀ ⇀ Problem: E and J are difficult to measure ⇀ ⇀ Solution: from E , we can get V , and from J , we can get I With J and E uniform inside   V I =ρ ∼ (14) L A V ρL = ∼ (15) I A Lecture 09 Angel Ramoso 24/ 29 Resistance For a conductor of resistivity ρ, ⇀ ⇀ E =ρ J ∼ (13) ⇀ ⇀ Problem: E and J are difficult to measure ⇀ ⇀ Solution: from E , we can get V , and from J , we can get I With J and E uniform inside   V I =ρ ∼ (14) L A V ρL = ∼ (15) I A The quantity ρL/A is called resis- tance (R). Lecture 09 Angel Ramoso 24/ 29 Resistance ˆ For a conductor with resistivity ρ, length L, and cross-sectional area A, Lecture 09 Angel Ramoso 25/ 29 Resistance ˆ For a conductor with resistivity ρ, length L, and cross-sectional area A, ρL R= ∼ (16) A Lecture 09 Angel Ramoso 25/ 29 Resistance ˆ For a conductor with resistivity ρ, length L, and cross-sectional area A, ρL R= ∼ (16) A ˆ SI unit: Volt/Ampere ( [V/A]) or Ohm ( [Ω]). Lecture 09 Angel Ramoso 25/ 29 Resistance ˆ For a conductor with resistivity ρ, length L, and cross-sectional area A, ρL R= ∼ (16) A ˆ SI unit: Volt/Ampere ( [V/A]) or Ohm ( [Ω]). ˆ If a conductor has a constant resistivity ρ, then R is just geometry-dependent. Lecture 09 Angel Ramoso 25/ 29 Resistance ˆ For a conductor with resistivity ρ, length L, and cross-sectional area A, ρL R= ∼ (16) A ˆ SI unit: Volt/Ampere ( [V/A]) or Ohm ( [Ω]). ˆ If a conductor has a constant resistivity ρ, then R is just geometry-dependent. ˆ Ohm’s Law (for constant R) Lecture 09 Angel Ramoso 25/ 29 Resistance ˆ For a conductor with resistivity ρ, length L, and cross-sectional area A, ρL R= ∼ (16) A ˆ SI unit: Volt/Ampere ( [V/A]) or Ohm ( [Ω]). ˆ If a conductor has a constant resistivity ρ, then R is just geometry-dependent. ˆ Ohm’s Law (for constant R) V = IR ∼ (17) Lecture 09 Angel Ramoso 25/ 29 Resistance ˆ For a conductor with resistivity ρ, length L, and cross-sectional area A, ρL R= ∼ (16) A ˆ SI unit: Volt/Ampere ( [V/A]) or Ohm ( [Ω]). ˆ If a conductor has a constant resistivity ρ, then R is just geometry-dependent. ˆ Ohm’s Law (for constant R) V = IR ∼ (17) ˆ Note: Ohm’s Law is only obeyed at constant ρ. Lecture 09 Angel Ramoso 25/ 29 Resistance Ohmic vs Nonohmic materials R is constant. R is not constant. Lecture 09 Angel Ramoso 26/ 29 Resistance Exercise Lecture 09 Angel Ramoso 27/ 29 Resistance Exercise An 18-gauge copper wire (ρcopper = 1.72×10−8 [Ω · m]) has a cross- sectional area of 8.2 × 10−7 m2. It carries a current of 1.67 [A].   Find: a. the electric field magnitude of the wire, b. the potential difference between two points in the wire 50 [m] apart, and c. The resistance of the wire with length 50 [m]. Lecture 09 Angel Ramoso 27/ 29 Resistance Solution Lecture 09 Angel Ramoso 28/ 29 Resistance Solution Electric Field Lecture 09 Angel Ramoso 28/ 29 Resistance Solution Electric Field E = ρJ ∼ (18) Lecture 09 Angel Ramoso 28/ 29 Resistance Solution Electric Field E = ρJ ∼ (18) ρI E= ∼ (19) A Lecture 09 Angel Ramoso 28/ 29 Resistance Solution Electric Field E = ρJ ∼ (18) ρI E= ∼ (19) A   V E = 0.03502926829 ∼ (20) m Lecture 09 Angel Ramoso 28/ 29 Resistance Solution Electric Field E = ρJ ∼ (18) ρI E= ∼ (19) A   V E = 0.03502926829 ∼ (20) m Potential Difference Lecture 09 Angel Ramoso 28/ 29 Resistance Solution Electric Field E = ρJ ∼ (18) ρI E= ∼ (19) A   V E = 0.03502926829 ∼ (20) m Potential Difference V = EL ∼ (21) Lecture 09 Angel Ramoso 28/ 29 Resistance Solution Electric Field E = ρJ ∼ (18) ρI E= ∼ (19) A   V E = 0.03502926829 ∼ (20) m Potential Difference V = EL ∼ (21) V = 1.751463415 [V] ∼ (22) Lecture 09 Angel Ramoso 28/ 29 Resistance Solution Resistance Lecture 09 Angel Ramoso 29/ 29 Resistance Solution Resistance ρL R= ∼ (23) A Lecture 09 Angel Ramoso 29/ 29 Resistance Solution Resistance ρL R= ∼ (23) A R = 1.04878 [Ω] ∼ (24) Lecture 09 Angel Ramoso 29/ 29 Resistance Solution Resistance ρL R= ∼ (23) A R = 1.04878 [Ω] ∼ (24) or alternatively, Lecture 09 Angel Ramoso 29/ 29 Resistance Solution Resistance ρL R= ∼ (23) A R = 1.04878 [Ω] ∼ (24) or alternatively, V R= ∼ (25) I Lecture 09 Angel Ramoso 29/ 29 Resistance Solution Resistance ρL R= ∼ (23) A R = 1.04878 [Ω] ∼ (24) or alternatively, V R= ∼ (25) I R = 1.04878 [Ω] ∼ (26) Lecture 09 Angel Ramoso 29/ 29