L08 Gradient Post

7/30/2019 L08 Gradient Post

1/19

LECTURE 8 slide 1

Lecture 8

Electrostatic Field and Potential

Gradient

Sections: 4.5, 4.6

Homework: D4.6, D4.7, D4.8; 4.16, 4.17, 4.18, 4.19, 4.20,

4.22, 4.23, 4.24, 4.25


2/19

LECTURE 8 slide 2

Conservative Property of Potential of Point Charge 1

the potential of a single point charge at the origin depends solelyon the radial distance to the observation point A (see L07)

1

4A

A

QV

r=

the potential difference VABbetween pointsA and B depends

solely on the their radial distances from the origin

N2

1 1

44

B

A

rB

AB r rBA r d

Q Q

V d dr r rr

= = = LE L a a

angular positions, and , of observation

points do not matter

+ r

E

B

Brpath of integration does not matter

integrand has only rcomponent and r

dependence


3/19

LECTURE 8 slide 3

Conservative Property of Potential of Point Charge 2

1 10

4

AA

A Ac

QV d

r r

= = =

E Lv

if path of integration is closed potential difference is zero

+ Br

E

c

vector field whose closed-path integral is zero for any closed

contour is called conservative


4/19

LECTURE 8 slide 4

Superposition of Potential

potential of discrete (point) charges

0 1

1( )

4 | |

Nn

nn

QV

==

r

r r

this is an algebraic superposition

potential due to distributed charge

0

1 ( )( ) , V

4 | |

vP

v

V V dv

= =

r

rr r

0

1 ( )( ) , V4 | |

s

s

V ds

= rr r r 0

1 ( )( ) , V4 | |

l

L

V dl

= rr r r

x

y

z

r

P

rQ

v


5/19

LECTURE 8 slide 5

Conservative Property of Electrostatic Potential

conservative property of potential follows from superposition and

conservative property of potential of point charge

if work along a closed path is zero for a single point charge, it will

be zero for any collection of charges

electrostatic potential taken on a closed integration path is zero

0AAc

V d= = E Lv

it follows that neither absolute potential nor

voltage depend on the path taken

x

y

zB

path #1

path#2#1 #2 #1 #2 0AA B BA AB ABV V V V V = + = =

#1 #2AB ABV V =


6/19

LECTURE 8 slide 6

Conservative Property and KVL

Kirchhoffs voltage law in circuits is a direct consequence of theconservative property of the electrostatic field

0 0nnc

d V = = E Lv

along any closed contour of a circuit the sum of the branch

voltages is zero


7/19

LECTURE 8 slide 7

Electrostatic Potential Gradient 1

consider a sufficiently small line element LAB along which E isconstant

( )

( ) ( )

AB AB A B B A

x y y z z x y z

V V V V V V

V E E E x y z

= = = =

= + + + +

E L

a a a a a a

x y zV E x E y E z = + +

forLAB0

( )x y zdV E dx E dy E dz = + +

on the other hand

V V VdV dx dy dz x y z

= + +

, ,x y zV V V

E E Ex y z

= = =

B x y zx y z = + + L a a a


8/19

LECTURE 8 slide 8

Electrostatic Potential Gradient 2

, ,x y z x y zV V V V V V

E E Ex y z x y z

= = = = + + E a a a

remember the vector operator from L06

x y xx y z

= + +

a a a

, V/mV =

E gradV V

the electric field vector equals the gradient of the potential with a

minus sign

the gradient ofV


9/19

LECTURE 8 slide 9

The distance between the plates of a parallel-plate capacitoris 10 mm: the left plate is at x = 0 and the right plate is at x =

10 mm. The left plate is at potential VL = 0 V and the right

one is at potential VR = 10 V. Find V(x) inside the capacitor

bearing in mind that E is constant.

Find E inside the capacitor.

x0LV =

10RV =

V

10 mm0


10/19

LECTURE 8 slide 10

Gradient and Directional Derivative 1

assume a scalar field V(x,y,z)

an infinitesimal displacement along the x-axis dxbrings us

to a slightly different scalar value V(x+dx,y,z)

( , , ) ( , , )x xd V V x dx y z V x y z d dx= + =L a6

( , , ) ( , , )y yd V V x y dy z V x y z d dy= + =L a6

( , , ) ( , , )z zd V V x y z dz V x y z d dz = + =L a6

there are respective changes in Vfor displacements dy and dz

an infinitesimal displacement y zd dx dy dz = + +L a a ainvokes all three changes at the same time

( , , ) ( , , )x y zdV d V d V d V V x dx y dy z dz V x y z = + + = + + +


11/19

LECTURE 8 slide 11


x y zV V VdV d V d V d V dx dy dz x y z

= + + = + +

( )x y z x y z

V V VdV dx dy dz

x y z

= + + + +

a a a a a a

LdV V d V dL = = L a

cos cos cos

x y z

L

dV V dx V dy V dz

dL x dL y dL z dL

dV V V V dL x y z

V

= + +

= + +

= a

Alternatively,

directional derivative

L

dVV

dL= a

x

y

zLa

x y

z

directional cosines


12/19

LECTURE 8 slide 12


LdV VdL

= a

the directional derivative shows the rate of change of the scalar

function in a specified direction aL

max min

| |, | |dV dV V VdL dL

= =

the maximum possible directional derivative is the magnitude of

the gradient itself

the gradient shows the direction and the magnitude of the

maximum rate of ascent of a scalar function

the directional derivative in any direction is determined by the

projection of the gradient onto this direction


13/19

LECTURE 8 slide 13

Gradient and E-field

V= E

the electrostatic field Epoints in the direction of the fastest

descent of the electrostatic potential and is equal to the rate of thisdescent


14/19

LECTURE 8 slide 14

Equipotential Surface

equipotential surface is the geometrical place of all points withequal potential

in 2-D problems the equipotential surface collapses into a line


15/19

LECTURE 8 slide 15

Equipotential Surface and Gradient

any direction tangential to the equipotential surface is a directionof zero directional derivative (no ascent/descent)

the directions normal to the

equipotential surface are the

directions of fastest ascent/descent, i.e., they are aligned with

gradV

1 VV =

2 VV =

4 VV =

3 VV =ta

naE

V

E

V

V

E

Can you estimate roughly the E fielddirection and magnitude from a

potential map?

0V

V V

= =

a a

V= E


16/19

LECTURE 8 slide 16

Equipotential Surface and Gradient

if a surface is defined by

( , , )V x y z const =

its unit normal can be found at any point as

| |

n

V

V

=

a


17/19

LECTURE 8 slide 17

Gradient in CCS and SCS

2 3

1 2 3

l

V V VdV dl dl dl V d

l l l

= + + =

L

N N N

1 2 3

1 1 1 2 2 2 3 3 3

dl dl dl

d h d h d h d = + +L a a a

1 2 3

1 2 3

V V VV

l l l

= + +

a a a

1CCS:

1 1

SCS: sin

z

r

V V VV

z

V V V

V r r r

= + +

= + +

a a a

a a a


18/19

LECTURE 8 slide 18

An equipotential surface is defined by V0 = 10. IfV(x,y,z) = y,

find: (a) the equation of the surface f(x,y,z) = 0; (b) the unit

normal an to the equipotential surface pointing away from the

origin; (c) the electric field vectorE on that side of thesurface which does not contain the origin.

Homework:


19/19

LECTURE 8 slide 19

You have learned:

that the electrostatic field is conservative, i.e., work does notdepend on the path taken and work is zero along a closed path

what gradient is and what directional derivative is

how to apply the principle of superposition in order to find the

potential of a system of charges

how to find the unit normal to a surface

that the E field is equal to the gradient of the potential Vwith a

minus sign

Documents

L08 Gradient Post