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Electric Field of a Continuous Charge Distribution

I. Electricity: Lecture 4 | 100:12 min

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Post by Daniel Brook on February 21, 2011

The Y-component aspect of the continuous charge distribution at the perpendicular bisector was really hard to understand. All of the trigonometry became really confusing. Either a different method of solving would have been nice or a more in-depth explanation of the problem could have helped.

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Last reply by: Kyle Kosic
Mon Apr 4, 2011 8:42 PM

Post by Kyle Kosic on April 4, 2011

When he has 1/a^2 * sin(theta), couldn't he replace sin(theta) with x/a?

Electric Field of a Continuous Charge Distribution

We know that the E-field produced by a point charge q at a point P a distance r away is (kq/r^2)*r_hat, where r_hat is a unit vector along the direction from q to P. How can we use this knowledge to find the E-field for a charge distribution?
The answer is the following: Divide the charge into many, many small elements, each element being so small that it is essentially a point. Then, the E-field produced by each element dq is obtained by using the formula for the field of a point charge. The total E-field is obtained by adding all the fields produced by all the charge elements; in other words, the total E-field is the integral over the charge distribution of (kdq/r^2)*r_hat.
Examples are given that show how to calculate the E-field produced by a charged rod, a charged ring, and a charged disk. In all these cases, the main issue is to set up the integral correctly, beyond which it becomes a problem in the calculus of integrations. In setting up the integral, we exploit the symmetry of the charge distribution to simplify the integral expression.

Electric Field of a Continuous Charge Distribution

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

AP Physics C: Electricity and Magnetism

I. Electricity
	Electric Force			56:18
		Intro		0:00
		Electric Charge		0:18
			Matter Consists of Atom	1:01
			Two Types of Particles: Protons & Neutrons	1:48
			Object with Excess Electrons: Negatively Charged	7:58
			Carbon Atom	8:30
			Positively Charged Object	9:55
		Electric Charge		10:07
			Rubber Rod Rubs Against Fur (Negative Charge)	10:16
			Glass Rod Rub Against Silk (Positive Charge)	11:48
			Hanging Rubber Rod	12:44
		Conductors and Insulators		16:00
			Electrons Close to Nucleus	18:34
			Conductors Have Mobile Charge	21:30
			Insulators: No Moving Electrons	23:06
			Copper Wire Connected to Excess Negative charge	23:22
			Other End Connected to Excess Positive Charge	24:09
		Charging a Metal Object		27:25
			By Contact	28:05
			Metal Sphere on an Insulating Stand	28:16
			Charging by Induction	30:59
			Negative Rubber Rod	31:26
			Size of Atom	36:08
		Extra Example 1: Three Metallic Objects		7:32
		Extra Example 2: Rubber Rod and Two Metal Spheres		6:25
	Coulomb's Law			87:18
		Intro		0:00
		Coulomb's Law		0:59
			Two Point Charges by Distance R	1:11
			Permitivity of Free Space	5:28
		Charges on the Vertices of a Triangle		8:00
			3 Charges on Vertices of Right Triangle	8:29
			Charge of 4, -5 and -2 micro-Coulombs	10:00
			Force Acting on Each Charge	10:58
		Charges on a Line		21:29
			2 Charges on X-Axis	22:40
			Where Should Q should be Placed, Net Force =0	23:23
		Two Small Spheres Attached to String		31:08
			Adding Some Charge	32:03
			Equilibrium Net Force on Each Sphere = 0	33:38
		Simple Harmonic Motion of Point Charge		37:40
			Two Charges on Y-Axis	37:55
			Charge is Attracted	39:52
			Magnitude of Net Force on Q	42:23
		Extra Example 1: Vertices of Triangle		9:39
		Extra Example 2: Tension in String		11:46
		Extra Example 3: Two Conducting Spheres		6:29
		Extra Example 4: Force on Charge		9:21
	Electric Field			97:24
		Intro		0:00
		Definition of Electric Field		0:11
			Q1 Produces Electric Field	3:23
			Charges on a Conductor	4:26
		Field of a Point Charge		13:10
			Charge Point Between Two Fields	13:20
			Electric Field E=kq/r2	14:29
			Direction of the Charge Field	15:10
			Positive Charge, Field is Radially Out	15:45
		Field of a Collection of a Point Charge		19:40
			Two Charges Q1,Q2	19:56
			Q1 Positive, Electric Field is Radially Out	20:32
			Q2 is Negative, Electric Field is Radially Inward	20:46
			4 Charges are Equal	23:54
		Parallel Plate Capacitor		25:42
			Two Plates ,Separated by a Distance	26:44
			Fringe Effect	30:26
			E=Constant Between the Parallel Plate Capacitor	30:40
		Electric Field Lines		35:16
			Pictorial Representation of Electric Field	35:30
			Electric Lines are Tangent to the Vector	35:57
			Lines Start at Positive Charge, End on Negative Charge	41:24
			Parallel Line Proportional to Charge	45:51
			Lines Never Cross	46:00
		Conductors and Shielding		49:33
			Static Equilibrium	51:09
			No Net Moment of Charge	53:09
			Electric Field is Perpendicular to the Surface of Conductor	55:40
		Extra Example 1: Plastic Sphere Between Capacitor		8:46
		Extra Example 2: Electron Between Capacitor		11:52
		Extra Example 3: Zero Electric Field		10:44
		Extra Example 4: Dimensional Analysis		6:01
	Electric Field of a Continuous Charge Distribution			100:12
		Intro		0:00
		General Expression For E		0:16
			Magnitude of Electric Field	1:29
			Disk: Spread Charge Distribution	5:04
			Volume Contains Charges	6:16
		Charged Rod One Dimension		16:28
			Rod in X-Axis	17:00
			Charge Density	17:49
			Find Electric Field at Distance 'A'	19:05
		Charged Rod, Cont.		32:48
			Origin at Center, Extends From -L to +L	33:11
			Dividing Rod into Pieces	34:50
			Electric Field Produced At Point P	35:09
			Another Element	37:43
			'Y' Components of Electric Field	42:15
		Charged Ring		54:23
			Find Electric Field Above the Center	54:48
		Charged Disc		58:43
			Collection of Rings	59:10
		Example 1: Charged Disk		17:18
		Example 2: Semicircle with Charge		7:49
		Example 3: Charged Cylindrical Charge		13:53
	Gauss's Law			87:00
		Intro		0:00
		Electric Field Lines		0:11
			Magnitude of Field	2:04
			Unit Area and Unit Lines	2:59
			Number of Lines Passing Through the Unit	6:45
		Electic Flux: Constant E		6:51
			Field Lines Equally Spaced	7:10
			Area Perpendicular To Field Lines	7:46
			Electric Flux	8:36
			Area Perpendicular to Electric Lines	9:43
			Tilt the Area	10:58
			Flux of E Through Area	17:30
		Electric Flux: General Case		20:46
			Perpendicular at Different Directions	23:24
			Electric Field Given On a Patch	27:10
			Magnitude of Field	28:53
			Direction is Outward Normal	29:34
			Flux Through Patch	30:36
		Example		36:09
			Electric Field in Whole Space	37:16
			Sphere of Radius 'r'	37:30
			Flux Through Sphere	38:09
		Gauss's Law: Charge Outside		46:02
			Flux Through Radius Phase is Zero	50:09
			Outward normal 'n'	54:55
		Gauss's Law: Charge Enclosed		60:30
			Drawing Cones	60:51
		Example 1: Flux Through Square		7:08
		Example 2: Flux Through Cube		10:23
		Example 3: Flux Through Pyramid		5:01
	Application of Gauss's Law, Part 1			66:48
		Intro		0:00
		When is Gauss Law Useful?		0:18
			Need a Surface S	5:14
			Gaussian Surface	5:50
		Sphere of Charge		10:11
			Charge Density is Uniform	10:30
			Radius as 'A'	11:23
			Case 1: R>A	11:58
			Any Direction On Cone Is Same	20:28
			Case 2: R<A	25:15
			Point R Within the Surface	25:30
		Concentric Cavity		31:11
			Inside Circle and Outside Circle	31:48
			R>A	32:17
			R<B	36:40
		Radius Dependent Charge Density		37:39
			Sphere	38:09
			Total Charge: Q	39:46
			Spherical Shell	40:13
			Finding Electric Field R>A	42:36
			R<A	44:14
		Example 1: Charged Sphere		9:56
		Example 2: Charged Spherical Cavity		11:06
	Application of Gauss's Law, Part 2			79:19
		Intro		0:00
		Infinitely Long Line of Charge		0:13
			All Points Same Magnitude	5:02
			E is Perpendicular to Line	9:08
			Gauss's Law Cannot be Applied to Finite Length	15:50
		Infinitely Long Cylinder Of Charge		16:05
			Draw a Cylinder of Radius 'R'	16:36
			Line of Charge Along the Center	18:25
			R<A	18:39
			Electric Field of Special Direction	19:06
		Infinite Sheet of Charge		25:12
			Electric Field Above the Sheet	25:38
			Point is Above Height, Cylinder Intersects	26:29
			Curved Path	33:12
		Parallel Plate Capacitors		37:16
			Electric Field Between Sheets	39:16
		Conductors		41:55
			Adding Charge to Conductors	42:16
			In Electrostatic Equilibrium Charges Stop Moving	44:37
			Electric Field is Perpendicular to Surface	47:16
			Excess Charge Must Reside on Surface	47:38
		Example 1: Cylindrical Shell		7:45
		Example 2: Wire Surrounded by Shell		6:43
		Example 3: Sphere Surrounded by Spherical Shell		7:30
	Electric Potential, Part 1			86:57
		Intro		0:00
		Potential Difference Between Two Points		0:16
			Electric Field in Space By Stationary Charges	0:30
			Point Charge Moves From A to B	1:37
			Electric Field Exerts a Force	1:50
			Electric Potential Energy	5:34
			Work Done By External Agent	20:03
			Change in Potential Energy is Equal to Amount of Work Done	24:06
		Potential Difference in Uniform Electric Field		27:59
			Constant Electric Field	28:22
			Equipotential	40:22
		Parallel Plates		40:52
			Electric Field is Perpendicular to Plate	42:07
			Charge Released at A from Rest	49:00
		Motion of Charged Particle in a Uniform Electric Field		51:55
		Example 1: Work by Moving Electrons		3:45
		Example 2: Block and Spring		13:52
		Example 3: Particle on String		11:27
	Electric Potential, Part 2			91:50
		Intro		0:00
		Potential of a Point Charge		0:32
			Potential Difference Between A to B	1:25
			Draw a Circle	9:12
			Tangential to Sphere	9:33
			Moving Normally From Sphere	12:33
		Potential Energy of a Collection of Charges		26:33
			Potential Energy of Two Charges	26:44
			Work Done in Assembling the Configuration	27:29
			Bringing From Infinity to New Location	33:57
			Work Done by External Agent	36:22
			Potential Energy of the System	39:39
			Potential Energy for Two Charges	40:00
		Example		44:49
			Two Charges	45:03
			Speed at Infinity	48:01
		Electric Field from the Potential		51:12
			Finding E if V is Given	51:33
		Electric Dipole		56:22
			Two Equal and Opposite Charges Separated By a Distance	56:32
			If a << r1 or r2	60:23
		Example 1: Two Point Charges		17:56
		Example 2: Two Insulating Spheres		7:31
		Example 3: Electric Potential of Space		4:01
	Electric Potential, Part 3			69:12
		Intro		0:00
			Continuous Charge Distribution	0:27
			Finding Potential for a Charge Point	1:39
			Potential Produced at P	4:42
		Charged Ring		8:38
			Electric Field at Some Point of Axis	9:13
		Charged Disk		19:32
			Collection of Ring	20:40
			Finding Potential Point Above the Ring	22:19
			Potential Due to The Ring	23:40
		Finite Line of Charge		35:56
			Line of Change Along the X-Axis and Y-axis	36:11
		Example 1: Charged Rod		8:52
		Example 2: Bent Semicircle		4:48
		Example 3: Bent Semicircle with Variables		4:52
	Electric Potential, Part 4			71:16
		Intro		0:00
		Charged Conductors		0:12
			Adding Excess Charge to a Conductor	1:02
			E=0 Inside Conductors	1:50
			Excess Charges Must Reside on Surface	3:40
			E Normal on the Surface	9:31
			Surface of Conductor is Equipotential	11:59
		Conducting Sphere		19:28
			Adding Charge to the Sphere	19:41
			Electric Field Outside is Concentrated at Center	20:05
			Electric Potential is Same as Center	23:01
		Example		26:24
			Two Spheres with Distance and of Different Size	26:45
			Connecting Both Spheres with Conducting Wire	27:22
		Cavity Within a Conductor		39:43
			Hollow Conductor	40:19
			Electric Static Equilibrium	41:13
			Electric Field is Zero Within Cavity	53:20
		Example 1: Neutral Conducting Sphere		4:03
		Example 2: Conducting Sphere with Spherical Shell		13:45
	Capacitor			84:14
		Intro		0:00
		Capacitance		0:09
			Consider Two Conductor s	0:25
			Electric Field Passing from Positive to Negative	1:19
			Potential Difference	3:31
			Defining Capacitance	3:51
		Parallel Plate Capacitance		8:30
			Two Metallic Plates of Area 'a' and Distance 'd'	8:46
			Potential Difference between Plates	13:12
		Capacitance with a Dielectric		22:14
			Applying Electric Field to a Capacitor	22:44
			Dielectric	30:32
		Example		34:56
			Empty Capacitor	35:12
			Connecting Capacitor to a Battery	35:26
			Inserting Dielectric Between Plates	39:02
		Energy of a Charged Capacitor		43:01
			Work Done in Moving a Charge, Difference in Potential	47:48
		Example		54:10
			Parallel Plate Capacitor	54:22
			Connect and Disconnect the Battery	55:27
			Calculating Q=cv	55:50
			Withdraw Mica Sheet	56:49
			Word Done in Withdrawing the Mica	60:23
		Extra Example 1: Parallel Plate Capacitor		8:41
		Extra Example 2: Mica Dielectric		15:01
	Combination of Capacitors			63:23
		Intro		0:00
		Parallel Combination		0:20
			Two Capacitors in Parallel With a Battery	0:40
			Electric Field is Outside	5:47
			Point A is Directly Connected to Positive Terminal	7:57
			Point B is Directly Connected to Negative Terminal	8:10
			Voltage Across Capacitor	12:54
			Energy Stored	14:52
		Series Combination		17:58
			Two Capacitors Connected End to End With a Battery	18:10
			Equivalent Capacitor	25:20
			A is Same Potential	26:59
			C is Same Potential	27:06
			Potential Difference Across First Capacitor (Va-Vb)	27:42
			(Vb-Vc) is Potential Difference Across Second Capacitor	28:10
			Energy Stored in C1,C2	29:53
		Example		31:07
			Two Capacitor in Series, 2 in Parallel, 3 in Parallel, 1 Capacitor Connected	31:28
			Final Equivalent Circuit	37:31
		Extra Example 1: Four Capacitors		16:50
		Extra Example 2: Circuit with Switches		8:25
	Calculating Capacitance			55:14
		Intro		0:00
		Considering a Sphere		0:28
			Placing Charge on Sphere	2:14
			On the Surface of Sphere	4:12
		Spherical Capacitor		9:20
			Sphere of Radius a and Shell of Radius b	9:40
			Positive Charge on Outer Sphere	11:02
			Negative Charge on Inner Sphere	11:26
			Calculating Potential Difference	11:38
		Parallel Plate Capacitor		22:38
			Two Plates with Charges Positive and Negative	22:54
			Separation of Plate	25:10
		Cylindrical Capacitor		28:40
			Inner Cylinder and Outer Cylindrical Shell	29:01
			Linear Charge Density	30:41
		Example 1: Parallel Plate Capacitor		4:39
		Example 2: Spherical Capacitor		8:51
	More on Filled Capacitors			77:13
		Intro		0:00
		Electric Dipole is an Electric Field : Torque		0:13
			Magnitude of Dipole	1:15
			Starts to Rotate	5:38
			Force qe to the Right	5:59
			Finding the Torque	6:35
		Electric Dipole is an Electric Field : Potential Energy		13:56
			Electric Field Try's to Rotate	14:43
			Object on Center of Earth	16:04
			Applying Torque Equal and Opposite	17:05
		Water Molecule		25:43
			Carbon Molecules	31:39
			Net Dipole Moment is Zero	32:11
			Induced Dipole Moment	34:43
		Filled Capacitor		35:27
			Empty Capacitor with Charge on it	35:44
			Inserting a Dielectric	36:08
		Capacitor Partially Filled with Metallic Slab		44:33
			Capacitor with Slab of Distance 'd'	44:54
		Capacitor Partially Filled with a Dielectric Slab		51:59
			Change in Potential Difference	53:28
		Example 1: Parallel Plate Capacitor		13:37
		Example 2: Conducting Slab		8:20
	Electric Current			79:17
		Intro		0:00
		Definition		0:20
			Consider a Wire ,Cylindrical	0:40
			Cross Sectional Area	1:06
			Crossing Charges Will be Counted	2:50
			Amount of Charge Crosses Cross Sectional Area	3:29
			Current I=q/t	4:18
			Charges Flowing in Opposite Direction	5:58
			Current Density	6:19
			Applying Electric Field	11:50
		Current in a Wire		15:24
			Wire With a Cross Section Area 'A'	15:33
			Current Flowing to Right	18:57
			How Much Charge Crosses Area 'A'	19:15
			Drift Velocity	20:02
			Carriers in Cylinder	22:40
		Ohm's Law		24:58
			Va-Vb = Electric Field times Length of Wire	28:27
			Ohm's Law	28:54
			Consider a Copper Wire of 1m , Cross Sectional Area 1cm/sq	34:24
		Temperature Effect		37:07
			Heating a Wire	37:05
			Temperature Co-Efficient of Resistivity	39:57
		Battery EMF		43:00
			Connecting a Resistance to Battery	44:30
			Potential Difference at Terminal of Battery	45:15
		Power		53:30
			Battery Connected with a Resistance	53:47
			Work Done on Charge	56:55
			Energy Lost Per Second	60:35
		Extra Example 1: Current		9:46
		Extra Example 2: Water Heater		8:05
	Circuits			94:08
		Intro		0:00
		Simple Rules		0:16
			Resistance in Series	0:33
			Current Passing Per Second is Equal	1:36
			Potential Difference	3:10
			Parallel Circuit, R1, R2	5:08
			Battery, Current Starts From Positive Terminal to Negative Terminal	10:08
		Series Combination of Resistances		13:06
			R1, R2 Connected to Battery	13:35
			Va-Vb=Ir1,Vb-Vc=Ir2	16:59
			Three Resistance Connected in Series Req=r1+r2+r3	18:55
		Parallel Combination of Resistance		19:28
			R1 and R2 Combined Parallel	19:50
			I=i1+i2 (Total Current)	24:26
			Requ=I/E	24:51
		A Simple Circuit		27:57
			Current Splits	29:15
			Total Resistance	31:52
			Current I= 6/17.2	35:10
		Another Simple Circuit		37:46
			Battery has Small Internal Resistance	38:02
			2 Ohms Internal Resistance, and Two Resistance in Parallel	38:24
			Drawing Circuit	48:53
			Finding Current	52:06
		RC Circuit		55:17
			Battery , Resistance and Capacitance Connected	55:30
			Current is Function of Time	58:00
			R, C are Time Constants	59:25
		Extra Example 1: Resistor Current/Power		4:17
		Extra Example 2: Find Current		6:03
		Extra Example 3: Find Current		10:00
		Extra Example 4: Find Current		13:49
	Kirchhoff's Law			102:02
		Intro		0:00
		First Kirchhoff Rule		0:19
			Two Resistance Connected With a Battery	0:29
			Many Resistance	1:40
			Increase in Potential from A to B	4:46
			Charge Flowing from Higher Potential to Lower Potential	5:13
		Second Kirchhoff Rule		9:17
			Current Entering	9:27
			Total Current Arriving is Equal Current Leaving	13:20
		Example		14:10
			Battery 6 V, Resistance 20, 30 Ohms and Another Battery 4v	14:30
			Current Entering I2+I3	21:18
		Example 2		31:20
			2 Loop circuit with 6v and 12 v and Resistance, Find Current in Each Resistance	32:29
		Example 3		42:02
			Battery and Resistance in Loops	42:23
		Ammeters and Voltmeters		56:22
			Measuring Current is Introducing an Ammeter	56:35
			Connecting Voltmeter, High Resistance	57:31
		Extra Example 1: Find Current		18:47
		Extra Example 2: Find Current		13:35
		Extra Example 3: Find Current		10:23
	RC Circuits			80:35
		Intro		0:00
		Charging a Capacitor: Circuit Equation		0:09
			Circuit with a Resistance , Capacitance and a Battery	0:20
			Closing Switch at T=0	1:36
			Applying Kirchhoff's Rule	6:26
			Change in Potential is Zero	6:52
			Solution Tau dq/dt= ec-q	16:25
		Discharging a Capacitor		27:14
			Charged Capacitor Connect to Switch and Resistance	27:30
			Closing the Switch at T=0	28:11
		Example		36:50
			12V Battery with Switch and Resistance 10mili ohms and Capacitor Connected 10 Micro Farad	37:02
			Time Constant	38:58
			Charge at q=0 at t=1sec	40:16
		Example		42:58
			Switch With Capacitor and Resistance	43:31
			What Time Charge C Has Initial Valve	45:17
			How Long Charge Energy Stored in C to Drop Half of Initial Value	46:55
		Example 1: RC Circuit 1		6:49
		Example 2: RC Circuit 2		12:53
		Example 3: RC Circuit 3		10:42
II. Magnetism
	Magnetic Field			98:19
		Intro		0:00
		Magnets		0:13
			Compass Will Always Point North	3:49
			Moving a Compass Needle	5:50
		Force on a Charged Particles		10:37
			Electric Field and Charge Particle Q	10:48
			Charge is Positive Force	11:11
			Charge Particle is At Rest	13:38
			Taking a Charged Particle and Moving to Right	16:15
			Using Right Hand Rule	23:37
			C= Magnitude of A, B	26:30
			Magnitude of C	26:55
		Motion of Particle in Uniform Magnetic Field		33:30
			Magnetic Field has Same Direction	34:02
			Direction of Force	38:40
			Work Done By Force=0	41:40
			Force is Perpendicular With Velocity	42:00
		Bending an Electron Beam		48:09
			Heating a Filament	48:29
			Kinetic Energy of Battery	51:54
			Introducing Magnetic Field	52:10
		Velocity Selector		53:45
			Selecting Particles of Specific Velocity	54:00
			Parallel Plate Capacitor	54:30
			Magnetic Force	56:20
			Magnitude of Force	56:45
		Extra Example 1: Vectors		19:24
		Extra Example 2: Proton in Magnetic Field		8:33
		Extra Example 3: Proton Circular Path		10:46
	Magnetic Force on a Current Carrying Conductor			64:43
		Intro		0:00
		Current Carrying Conductor in a Magnetic Field		0:19
			Current Though the Wire Connected to Battery	1:22
			Current Exerts Force Toward the Left	2:16
			IF Current is Reversed ,Force Exerts on Right	2:47
		Magnetic Force		3:31
			Wire with Current 'I' and with magnetic Field	4:02
			Force Exerted by Magnetic field	5:05
			Applying right hand Rule	5:25
			Let N be Number of Charge Carries Per /Vol	6:40
			Force on Wire	8:30
			Number of Charge Crossing in Time 't'	12:51
		Example		22:32
			Wire Bent to Semi Circle and Rest is Straight	22:51
			Applying Constant Magnetic Field in 'y' Direction	23:24
			Force n Straight Segment	23:50
			Net Force	34:19
		Example 1: Rod on Rails		15:37
		Example 2: Magnetic Force on Wire		13:59
	Torque on a Current Carrying Loop			69:06
		Intro		0:00
		B-Field Parallel to Plane of the Loop		0:27
			Loop in the X-Y Plane	1:06
			Net Force on Loop	7:45
		B-Field Not Parallel to Plane of the Loop		15:16
			Loop in the X-Y Plane, Free to Rotate in X- Direction	15:32
			Force on Out of Page and Force in to the Page	15:59
			Loop Turns Through 90 Degrees	18:10
		Magnetic Moment		36:26
			Any Current Loop Has Current 'I'	36:51
			Electric Dipole in Electric Field	38:17
			Potential Energy	39:54
			Magnetic Potential Energy of Dipole	41:05
		Example		43:33
			Circular of Radius 'r' With Magnetic Field and Pass Current	43:42
			Torque	46:01
		Example 1: Loop in Magnetic Field		9:21
		Example 2: Rotating Charge		10:32
	Magnetic Field Produced By Current, Part 1			57:58
		Intro		0:00
		Biot-Savart Law		0:11
			Suppose A current Carrying Wire	0:50
			Magnetic Field Produced by the Tiny Element is Also Tiny	3:09
			Permeability of Free Space	4:56
		B-Field of a Straight Wire		8:40
			Wire in X Axis	9:05
			What is the Magnetic Field Produce at Point p	9:16
			Taking a Small Segment	9:57
			If Length is Infinite	26:26
		Semi Circular Wire		27:02
			Semicircular Wire of Radius 'R'	27:22
			Finding Magnetic Field at Center	27:48
		Circular Current in Loop		33:37
			Circular Loop with Current 'I'	33:47
			Current Above the Center	34:00
		Example 1: Loop Carrying Current		10:42
		Example 2: Concentric Loops		4:57
	Magnetic Field Produced By Current, Part 2			79:29
		Intro		0:00
		Ampere's Law		0:16
			Consider a Loop at Any Point in Loop	1:15
		Long Cylindrical Wire		9:08
			Wire of Radius 'r'	9:24
			Magnetic Field is Tangent to Circle and Has Same Magnitude	10:15
			B at r>R	21:58
			B at r<R	23:08
			B at r=R	25:49
		Toroid		26:58
			Wrap a Wire to Toroid	27:47
			Calculating the Magnetic Field for 1 Loop	29:30
		Solenoid		39:17
			Coil With Many Turns	39:35
			Each Loop Carrying Current	40:29
			Taking Loop Within the Solenoid and Close the Loop	43:05
			Applying Ampere's Law	43:33
		Example 1: Infinitely Long Wire		8:12
		Example 2: Straight Wire		4:15
		Example 3: Two Parallel Conductors		8:21
		Example 4: Solenoid		10:13
	Magnetic Field Produced By Current, Part 3			50:37
		Intro		0:00
		Magnetic Force Between Parallel Conductors		0:16
			Two Parallel Plate Capacitors with Current	0:40
			Magnetic Field by i1	1:50
			According to Right Hand Rule	2:37
		Example		10:20
			Wire of 4m Length	10:50
			Mass of Wire 1Kg	11:18
			Force of Repulsion =Mg	12:24
		Gauss's Law in Magnetism		15:36
			Surface of Area, Magnetic Field is Perpendicular to Surface	17:09
			Magnetic Flux Through Enclosed surface	19:23
		Example		26:44
			Magnetic Field Out of Page	27:54
			Consider a Flux Through Rectangular Loop	28:52
		Example 1: Two Parallel Wires		9:45
		Example 2: Cube with Magnetic Field		5:36
	Faraday's Law			70:38
		Intro		0:00
		Faraday's Law		0:14
			Coil Connected to Ammeter	0:29
			Introducing a Magnet	1:08
			Moving the Magnet Forward and Backward	1:33
			Flux Increasing in Time	2:20
			Induced Electro Motive Force EMF	4:20
			Iron Core Square with Battery and Switch, Ammeter	5:22
			Close the Switch, Current Appears	6:11
		Lenz's Law		9:17
			Wire with Current I and Wire Loop	9:30
			Magnetic Field is Into the Page	10:14
			Current Induced in Wire to Oppose Change in Flux	12:54
			Example: Two Wires with Resistance and Uniform Magnetic Field	16:00
		Increasing B		29:02
			Coil of 100 Turns	29:20
			B Perpendicular to Coil	30:47
			Flux Through Each Turn	32:25
		Rotating Coil		37:36
			Consider a Big Magnet and Rectangular Coil with many Turns	37:49
			Rotating Coil With Angular Velocity 'w'	41:49
		Example 1: Loop		9:51
		Example 2: Solenoid		6:57
		Example 3: Wrapped Square		7:16
	Motional EMF			60:17
		Intro		0:00
		Moving a Conducting Rod in Magnetic Field		0:24
			Rod Moving in a Plane with Velocity 'v'	0:49
			Charges Piles Up and Down Until Electric Force Balance 'B'	7:59
			Equilibrium	9:30
			Potential Difference, Distance to Length of Wire	9:59
		Rod Pulled By External Agent		11:30
			Resistance to Wire	12:01
			Introducing Uniform Magnetic Field into The page	12:14
			Finding Flux	14:45
			Power Delivered to Resistance	17:01
			Force Exerted by 'B' on Rod	19:10
			Power By Agent	22:26
		Sliding Rod		23:08
			Resistance with a Sliding Rod and Magnetic Field 'B'	23:35
			Push With Initial Velocity 'V0'	24:01
			Finding Current = I	25:20
		Rotating Rod		36:10
			Magnetic Field into The Page	36:19
			Rod fixed in Plane and Rotating	36:40
			Induced EMF in Segment	40:00
		Example 1: Bar in Magnetic Field		6:15
		Example 2: Rod in Magnetic Field		11:08
	Induced Electric Field			65:19
		Intro		0:00
		Change B to Induce E		0:54
			Loop with Magnetic Field B	1:10
			Flux is Positive With Choice of 'n'	2:45
			Suppose Magnetic Field is Changing	3:04
			B Changing with time Flux (>0)	3:24
			Change in Electric Field Induces magnetic Field	20:34
		Example		21:08
			Cylinder with Magnetic Field	21:20
			Fill With Radius 'r'	22:11
			Turn Off the Field	22:30
			Magnetic Flux Through Big Loop	29:59
		AC Generator		38:28
			Magnetic Field with Coil of Many Turns	38:50
			As the Coil Rotates Flux is Induced	39:18
			Coil Rotated by Angle	40:29
			Coil Connected to The Ring and End Connected to Lamp	42:12
			Kinetic Energy Strike the Coil and Rotating Coil will Produce Electric Energy	45:12
		Example 1: Electric Field		12:09
		Example 2: Electric Field		7:00
	Inductance			71:10
		Intro		0:00
		Mutual Inductance		0:10
			Two Coils	0:35
			Current is Time Dependent	0:54
			Flux Proportional	1:55
			Magnetic Flux in Coil 2	2:08
			Induced EMF	2:40
			Flux Through 2nd Coil Proportional to Current in First Coil	4:07
			Mutual Inductance	5:30
			Suppose Current is in 2nd Coil	9:28
		Example		12:15
			Two Coils M=0.001	12:26
			Φ= Mi1	14:17
			Induced EMF	15:44
		Example		18:30
			Solenoid with N turns	18:40
			B inside Solenoid	21:05
			Φ Through the Ring	22:14
		Self Inductance		27:50
			Single Coil with Current	28:33
			I with Time Dependent	28:54
			Φ Proportional to B , Proportional to I	30:00
			Induced EMF =-di/dt	31:27
		Example 1: Circular Wire		15:46
		Example 2: Two Coils		9:54
		Example 3: Coil		7:24
	RL Circuits			85:19
		Intro		0:00
		Current Raising		0:45
			Battery and Switch with Resistance and Inductance	1:17
			Close s1 at T=0	2:27
			With out Inductor , Current is E/R	4:03
			I at T=0	9:51
			Vb-Va= -Ir	15:05
			Log (i-e/r)	19:51
		Current Declining		27:16
			Resistance R and Inductance	27:37
			I= E/R	28:37
			Switch is On at T=0	29:10
		Example		39:46
			Battery and Resistance R Connected with Inductor	39:55
			Time Constant l/R	40:58
			Time to Reach Half Time	41:59
			per τ (1-1/e)	44:36
		Magnetic Energy		45:47
			E-IR-Ldi/dt	46:26
			Power Derived By Current	46:51
			Magnetic Energy Stored in Conductor	52:48
			U=Li2	55:28
		Magnetic Energy Density		57:49
			Solenoid	58:18
			U=1/2 Li2	59:03
			Energy Density	60:45
		Example 1: Circuit 1		6:13
		Example 2: Circuit 2		16:54
	Circuit Oscillation			82:26
		Intro		0:00
		Oscillation in LC Circuit: Qualitative Analysis		0:30
			Circuit with Capacitance and Inductance	1:27
		Comparison with a Spring Block System		4:57
			Close the Switch, Let the Block Move	5:51
			At V=0	7:06
		LC Circuit Oscillation :Quantitative Analysis		15:07
			U Total = Ue + U m	17:26
		Example RLC		29:25
			Battery =12V, Capacitor and Inductor	29:54
			Switch at B F> t	31:42
			Damped Oscillation	50:14
		Example 1: LC Circuit 1		7:34
		Example 2: LC Circuit 2		16:19
		Example 3: RLC Circuit		6:52
	Maxwell's Equations			72:35
		Intro		0:00
		Displacement Current		1:29
			Ampere's Law	3:04
			Surface Bounded by Path	3:48
			I Current Going Through Surface	4:53
			Charging a Capacitor	9:55
		Maxwell's Equation		18:26
			Integral Form	18:53
			E.da =Q/e0 in Closed Surface	18:55
			Absence of Magnetic Monopoles	19:55
			Flux Through the Surface Bounded By C	22:26
			Ampere's Law	23:01
		Plane Electromagnetic Wave		31:03
			Electric and Magnetic Field	31:27
		Example		39:20
			Electromagnetic Wave Traveling in X Direction	39:40
			Lamda=c/f	41:30
			B=E/C	43:49
		Energy and Momentum Carried by EM Waves		44:34
			Energy Density	46:35
			Area in Y-Z Plane , Wave in X -Direction	48:53
			Energy Crossing Per Unit Area	52:53
			Pointing Vector	53:11
			Reflection of Radioactive	60:26
		Example 1: Cylindrical Region		8:36
		Example 2: Electric Field of EM Wave		3:16

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