8. Potential and potential energy and examples
pinterest Announcements: Important definitions:
Electric potential energy (units of energy: J, eV, …)
Electric potential, or potential, or voltage are all the same (units of potential: V, J/C)
Both are only defined up to a constant (like the gravitational energy, where I can define the zero of energy to be see level or the floor of Leacock 132)
Equipotential: (lines of same voltage)
Similar to topographic map equipotentials (lines of same altitude) https://phet.colorado.edu/sims/html/charges-and- fields/latest/charges-and-fields_en.html Potential (gravitational) energy and force:
푚 푚푀 퐺 U = − ⊙ 푟 Potential energy
www.s-cool.co.uk 푀⊙
푠푝푎푐푒
푈푠푝푎푐푒 − 푈푒푎푟푡ℎ = − න 퐹Ԧ ∙ 푑푙 푒푎푟푡ℎ Potential (gravitational) energy Potential (electric) energy and force: and force:
푚푀 퐺 푞푄 푘 U = − ⊙ U = 푒푛푐푙 0 푟 푟 Potential energy Electric potential energy
B
A
푠푝푎푐푒 퐵
푈푠푝푎푐푒 − 푈푒푎푟푡ℎ = − න 퐹Ԧ ∙ 푑푙 푈퐵 − 푈퐴 = − න 푞퐸 ∙ 푑푙 푒푎푟푡ℎ 퐴 Potential (electric) Potential (electric) energy and electric field: and force:
푄 푘 푞푄 푘 V = 푒푛푐푙 0 U = 푒푛푐푙 0 푟 푟 Electric potential between A and B Electric potential energy (or voltage between A and B) B B
A A
퐵 퐵
푉퐵 − 푉퐴 = − න 퐸 ∙ 푑푙 푈퐵 − 푈퐴 = − න 푞퐸 ∙ 푑푙 퐴 퐴 Example 0: potential of a uniform electric field E and V for an Infinite Sheet of Charge
The equipotential lines are the dashed blue lines. The electric field lines are the brown lines. The equipotential lines are everywhere perpendicular to the field lines.
퐵
푉퐵 − 푉퐴 = − න 퐸 ∙ 푑푙 = −퐸푑 퐴
d line integral (follow any line) VA VB
Section 25.4 Example 0: potential of a uniform electric field E and V for an Infinite Sheet of Charge
The equipotential lines are the dashed blue lines. The electric field lines are the brown lines. The equipotential lines are everywhere perpendicular to the field lines.
퐵
푉퐵 − 푉퐴 = − න 퐸 ∙ 푑푙 = −퐸푑 퐴
푉 − 푉 ⇒ 퐸 = − 퐵 퐴 d 푑 VA VB
Section 25.4 Example 0: potential of a uniform electric field E and V for an Infinite Sheet of Charge
The equipotential lines are the dashed blue lines. The electric field lines are the brown lines. The equipotential lines are everywhere perpendicular to the field lines.
퐵
푉퐵 − 푉퐴 = − න 퐸 ∙ 푑푙 = −퐸푑 퐴
휕푉 ⇒ 퐸 = − d 푥 휕푥 VA VB
Section 25.4 x In general:
휕푉 퐸 = − 푥 휕푥 퐵 휕푉 퐸 = − 푦 휕푦 ֞ 푉퐵 − 푉퐴 = − න 퐸 ∙ 푑푙 퐴 휕푉 퐸 = − 푧 휕푧 line integral In general:
휕푉 퐸 = − 푥 휕푥 퐵 휕푉 퐸 = − 푦 휕푦 ֞ 푉퐵 − 푉퐴 = − න 퐸 ∙ 푑푙 퐴 휕푉 퐸 = − 푧 휕푧 line integral
This is also called the gradient of V: 퐸 = −푔푟푎푑(푉) Example 1: potential of a point charge
An isolated positive point charge produces a field directed radially outward. The potential difference between points A and B will be
11 VB− V A = k e q − rrBA
Section 25.3 Example 1: potential of a point charge
The electric potential is independent of the path between points A and B.
It is customary to choose a reference potential of V = 0 at rA = ∞. Then the potential due to a point charge at some point r is q Vk= e r
Section 25.3 Example 2: potential of a spherical charge Van de Graaff Generator
Charge is delivered continuously to a high- potential electrode by means of a moving belt of insulating material. The high-voltage electrode is a hollow metal dome mounted on an insulated column. Large potentials can be developed by repeated trips of the belt. Protons accelerated through such large potentials receive enough energy to initiate nuclear reactions.
Section 25.8 What is the potential at the surface of the Van de Graaff generator?
A. 1-10V B. 10-100V C. 100-1kV D. 1kV-10kV E. 100kV-1MV F. 1MV-10MV How would you determine the potential at the surface of the Van de Graaff? Rank Responses 1 VOLTMETER 2 TOUCH IT 3 IDK 4 MATH 5 MASMIS 6 Other How would you determine the potential at the surface of the Van de Graaff? The Van der Graaff experiment to determine the electrostatic푽풂 potential 푸 d ퟎ 푽=0
Breakdown electric field in air is 푉 푉 퐸 ≅ 3푀 = 3 ∗ 106 퐵퐷 푚 푚 ∆푉 0 − 푉푎 (assuming a uniform = − = − electric field between 푑 푑 the two spheres)
6 푉푎 ≅ 3 ∗ 10 푑 Example 2: potential of a spherical charge
The Van der Graaff experiment to determine the electrostatic potential 푄0푘0푟Ƹ 푸ퟎ 퐸 = ⇒ Radius a 푟2 The Van der Graaff experiment to determine the electrostatic potential 푄0푘0푟Ƹ 푸ퟎ 퐸 = ⇒ Radius a 푟2
푏 푏 푄 푘 푟Ƹ 푉 −푉 = − න 퐸 ∙ 푑푟 = − න 0 0 ∙ 푑푟 푏 푎 푟2 푎 푎 The Van der Graaff experiment to determine the electrostatic potential 푄0푘0푟Ƹ 푸ퟎ 퐸 = ⇒ Radius a 푟2
푏 푏 푄 푘 푟Ƹ 푉 −푉 = − න 퐸 ∙ 푑푟 = − න 0 0 ∙ 푑푟 푏 푎 푟2 푎 푎 푏 푄 푘 푑푟 = − න 0 0 푟2 푎 The Van der Graaff experiment to determine the electrostatic potential 푄0푘0푟Ƹ 푸ퟎ 퐸 = ⇒ Radius a 푟2
푏 푏 푄 푘 푟Ƹ 푉 −푉 = − න 퐸 ∙ 푑푟 = − න 0 0 ∙ 푑푟 푏 푎 푟2 푎 푎 푏 푄 푘 푑푟 = − න 0 0 푟2 푎 푄 푘 푉 −푉 = 0 0]푏 푏 푎 푟 푎 The Van der Graaff experiment to determine the electrostatic potential 푄0푘0푟Ƹ 푸ퟎ 퐸 = ⇒ Radius a 푟2
푏 푏 푄 푘 푟Ƹ 푉 −푉 = − න 퐸 ∙ 푑푟 = − න 0 0 ∙ 푑푟 푏 푎 푟2 푎 푎 푏 푄 푘 푑푟 = − න 0 0 푟2 푎 푄 푘 푄 푘 푄 푘 푉 −푉 = 0 0]푏 = 0 0 − 0 0 푏 푎 푟 푎 푏 푎 Example 2: potential of a spherical charge
The Van der Graaff experiment to determine the electrostatic potential 푄0푘0푟Ƹ 푸ퟎ 퐸 = ⇒ Radius a 푟2
푏 푏 푄 푘 푟Ƹ 푉 −푉 = − න 퐸 ∙ 푑푟 = − න 0 0 ∙ 푑푟 푏 푎 푟2 푎 푎 푏 푄 푘 푑푟 = − න 0 0 푟2 푎 푄 푘 푄 푘 푄 푘 푉 −푉 = 0 0]푏 = 0 0 − 0 0 푏 푎 푟 푎 푏 푎 푄 푘 Assume 푏 = ∞ and 푉 = 0 ⇒ 푉 = 0 0 푏 푎 푎 Example 3: potential of multiple charges