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