physics ch 23 help

7/30/2019 Physics Ch 23 Help

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23.3 Calculating Electric Potential23.3 Calculating Electric Potential

Use two approaches in calculating the potential dueto a charge distribution:

1) if the charge distribution is known, we can use

or

2) if we know how the electric field depends onposition, we can use

(sometimes use a combination of these two approaches)


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Example: A charged conducting sphere

A solid conducting sphere of radius R has a total charge q. Find

the potential everywhere, both outside and inside the sphere.


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Example: Oppositely charged parallel plates

Find the potential at any height y between two oppositely

charged parallel plates.

usually take

potential at b to bezero


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Example: An infinite line charge or charged conductingcylinder

Find the potential at a distance r from a very long line chargewith linear charge density .

by setting Vb

= 0 at

point b at an arbitraryradial distance r

0


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Example: A ring of charge

Electric charge is distributed uniformly around a thin ring of

radius a, with total charge Q. Find the potential at a point P onthe ring axis at a distance x from the centre of the ring.


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Example: A line of charge

Electric charge Q is distributed uniformly along a line or thin

rod of length 2a. Find the potential at a point P along theperpendicular bisector of the rod at a distance x from its centre.


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Example: Infinite plane and point charge

Ans:


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23.4 Equipotential Surfaces23.4 Equipotential Surfaces

The potential at various points in an E-field can berepresented graphically by equipotential surfaces

these are three-dimensional surfaces on which theelectric potential Vis the same at every point

if q0 moves from point to point on this surface, the

electric potential energy q0Vremains constant since the potential energy does not change as q0

moves over an equipotential surface, Edoes no work

this implies that the E-field must be perpendicular to

the equipotential surface at every point


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Cross sections of equipotentialsurfaces (blue lines) andelectric field lines (red lines) fordifferent arrangements ofcharges


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Equipotentials and ConductorsEquipotentials and Conductors

When all charges are at

rest, the surface of aconductor is always anequipotential surface

When all charges are atrest, the E-field justoutside a conductor mustbe perpendicular to the

surface at every point


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We know E= 0 everywhere inside the conductor!

at any point just inside the surface the component of Etangent to the surface is zero

it follows that the tangential component of Eis also zerojust outside the surface

therefore Eis perpendicular to the surface at each point

consider q0

moving

around a rectangularloop and returning to

its starting point


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Theorem :

In an electrostatic situation, if a conductor contains a cavityand if no charge is present inside the cavity, then there canbe no net charge anywhere on the surface of the cavity.


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23.5 Potential Gradient23.5 Potential Gradient

From before:

if we know Ewe can calculate V

now we consider the reverse; if we know Vhow do wecalculate E?

To do this we now consider Vas a function of thecoordinates (x,y,z) of a point in space:


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Va

Vb

is the change of potential energy when a point

moves from b to a:

dVis the infinitesimal change of potentialaccompanying an infinitesimal elementdlof the path

from b to a

(i.e., integrands are equal)


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write Eand dlin terms of their components:

the components of Ecan be written in terms of V

in terms of unit vectors we can write Eas


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in vector notation the following operator is called thegradient of the function f:

therefore in vector notation and in terms of thegradient the E-field is given by

the potential gradient

if Eis radial with respect to a point and ris thedistance from the point then the above corresponds to


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Example: Field of a point charge

Find the vector electric field given that the potential at a radialdistance r from a point charge q is

Example: Field outside a charged conducting cylinder

Find the components of the electric field outside the cylindergiven that the potential outside the cylinder with radius R andcharge per unit length is

Example: Field of a ring charge

Find the electric field at a point P given that the potential atpoint P on the axis a distance x from the centre of a ring, withradius a and total charge Q, is

physics ch 23 help

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