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  • Explain point charges and express the equation for electric potential of a point charge.
  • Distinguish between electric potential and electric field.
  • Determine the electric potential of a point charge given charge and distance.

Point charges, such as electrons, are among the fundamental building blocks of matter. Furthermore, spherical charge distributions (like on a metal sphere) create external electric fields exactly like a point charge. The electric potential due to a point charge is, thus, a case we need to consider. Using calculus to find the work needed to move a test charge q size 12{q} {} from a large distance away to a distance of r size 12{r} {} from a point charge Q size 12{Q} {} , and noting the connection between work and potential W = q Δ V size 12{ left (W= - q?V right )} {} , it can be shown that the electric potential V size 12{V} {} of a point charge is

V = kQ r ( Point Charge ) , size 12{V= { { ital "kQ"} over {r} } \( "Point Charge" \) ,} {}

where k is a constant equal to 9.0 × 10 9 N · m 2 / C 2 .

Electric potential V size 12{V} {} Of a point charge

The electric potential V size 12{V} {} of a point charge is given by

V = kQ r ( Point Charge ) . size 12{V= { { ital "kQ"} over {r} } \( "Point Charge" \) ,} {}

The potential at infinity is chosen to be zero. Thus V size 12{V} {} for a point charge decreases with distance, whereas E size 12{E} {} for a point charge decreases with distance squared:

E = F q = kQ r 2 . size 12{ left (E=F/q right )= ital "kQ/r" rSup { size 8{2} } } {}

Recall that the electric potential V size 12{V} {} is a scalar and has no direction, whereas the electric field E size 12{E} {} is a vector. To find the voltage due to a combination of point charges, you add the individual voltages as numbers. To find the total electric field, you must add the individual fields as vectors , taking magnitude and direction into account. This is consistent with the fact that V size 12{V} {} is closely associated with energy, a scalar, whereas E size 12{E} {} is closely associated with force, a vector.

What voltage is produced by a small charge on a metal sphere?

Charges in static electricity are typically in the nanocoulomb nC size 12{ left ("nC" right )} {} to microcoulomb µC size 12{ left (µC right )} {} range. What is the voltage 5.00 cm away from the center of a 1-cm diameter metal sphere that has a −3.00 nC static charge?

Strategy

As we have discussed in Electric Charge and Electric Field , charge on a metal sphere spreads out uniformly and produces a field like that of a point charge located at its center. Thus we can find the voltage using the equation V = kQ / r size 12{V= ital "kQ"/r} {} .

Solution

Entering known values into the expression for the potential of a point charge, we obtain

V = k Q r = 8.99 × 10 9 N · m 2 / C 2 –3.00 × 10 –9 C 5.00 × 10 –2 m = –539 V. alignl { stack { size 12{V=k { {Q} over {r} } = left (9 "." "00" times "10" rSup { size 8{9} } " N" cdot m rSup { size 8{2} } /C rSup { size 8{2} } right ) { { - 3 "." "00" times "10" rSup { size 8{ - 9} } " C"} over {5 "." "00" times "10" rSup { size 8{"–2"} } " m"} } } {} #= - "540"" V" "." {} } } {}

Discussion

The negative value for voltage means a positive charge would be attracted from a larger distance, since the potential is lower (more negative) than at larger distances. Conversely, a negative charge would be repelled, as expected.

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What is the excess charge on a van de graaff generator

A demonstration Van de Graaff generator has a 25.0 cm diameter metal sphere that produces a voltage of 100 kV near its surface. (See [link] .) What excess charge resides on the sphere? (Assume that each numerical value here is shown with three significant figures.)

The figure shows a Van de Graaff generator. The generator consists of a flat belt running over two metal pulleys. One pulley is positioned at the top and another at the bottom. The upper pulley is surrounded by an aluminum sphere. The aluminum sphere has a diameter of twenty five centimeters. Inside the sphere, the upper pulley is connected to a conductor which in turn is connected to a voltmeter for measuring the potential on the sphere. The lower pulley is connected to a motor. When the motor is switched on, the lower pulley begins turning the flat belt. The Van de Graaff generator with the above described setup produces a voltage of one hundred kilovolts. The potential on the surface of the sphere will be the same as that of a point charge at the center which is twelve point five centimeters away from the center. Thus the excess charge is calculated using the formula Q equals r times V divided by k.
The voltage of this demonstration Van de Graaff generator is measured between the charged sphere and ground. Earth’s potential is taken to be zero as a reference. The potential of the charged conducting sphere is the same as that of an equal point charge at its center.

Strategy

The potential on the surface will be the same as that of a point charge at the center of the sphere, 12.5 cm away. (The radius of the sphere is 12.5 cm.) We can thus determine the excess charge using the equation

V = kQ r . size 12{V= ital "kQ"/r} {}

Solution

Solving for Q and entering known values gives

Q = rV k = 0 . 125 m 100 × 10 3 V 8.99 × 10 9 N · m 2 / C 2 = 1.39 × 10 –6 C = 1.39 µC. alignl { stack { size 12{Q= { { ital "rV"} over {k} } = { { left (0 "." "12"" m" right ) left ("100"´"10" rSup { size 8{3} } " V" right )} over {9 "." "00"´"10" rSup { size 8{9} } " N" cdot m rSup { size 8{2} } /C rSup { size 8{2} } } } } {} #=1 "." "39"´"10" rSup { size 8{-6} } " C=1" "." "39 "mC "." {} } } {}

Discussion

This is a relatively small charge, but it produces a rather large voltage. We have another indication here that it is difficult to store isolated charges.

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Questions & Answers

is there more then 4 dimensions
Miguel Reply
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princy
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Miguel
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Brown
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ayesha
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Brown
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ayesha Reply
it what?
Luke
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ayesha
the basic formula is uncertainty in momentum multiplied buy uncertainty In position is greater than or equal to 4×pi/2. same formula for energy and time
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Is there a formula for time of free fall given that the body has initial velocity? In other words, formula for time that takes a downward-shot projectile to hit the ground. Thanks!
Cyclone Reply
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Agboro
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Chandan
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Sahim
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Bile
Hy
Kulsum
What is unit of watt?
Kulsum
watt is the unit of power
Rahul
p=f.v
Rahul
watt can also be expressed as Nm/s
Rahul
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Maxamed
SI unit of mass is Kg(kilogram).
Robel
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Maxamed
Formula for for the falling body with initial velocity is:v^2=v(initial)^2+2*g*h
Mateo
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2 forces whose resultant is 100N, are at right angle to each other .if one of them makes an angle of 30 degree with the resultant determine it's magnitude
Victor Reply
50 N... (50 *1.732)N
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50 N...(50 *1.732)N
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Syed
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Syed
.
muhammad
Is earth is an inertial frame?
Sahim Reply
The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool that was in use in Europe, China and Russia, centuries before the adoption of the written Hindu–Arabic numeral system
Sahim
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Irungu
Most welcome
Sahim
Hey.. I've a question.
Sahim Reply
Is earth inertia frame?
Sahim
only the center
Shii
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Irungu
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prakash Reply
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prakash
a load of 20N on a wire of cross sectional area 8×10^-7m produces an extension of 10.4m. calculate the young modules of the material of the wire is of length 5m
Ebenezer Reply
Young's modulus = stress/strain strain = extension/length (x/l) stress = force/area (F/A) stress/strain is F l/A x
El
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Ebenezer
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Ebenezer
two bodies x and y start from rest and move with uniform acceleration of a and 4a respectively. if the bodies cover the same distance in terms of tx and ty what is the ratio of tx to ty
Oluwatola Reply
what is cesium atoms?
prakash Reply
The atoms which form the element Cesium are known as Cesium atoms.
Naman
A material that combines with and removes trace gases from vacuum tubes.
Shankar
what is difference between entropy and heat capacity
Varun
Heat capacity can be defined as the amount of thermal energy required to warm the sample by 1°C. entropy is the disorder of the system. heat capacity is high when the disorder is high.
Chathu
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What is realm
Vinodhini Reply
The quantum realm, also called the quantum scale, is a term of art inphysics referring to scales where quantum mechanical effects become important when studied as an isolated system. Typically, this means distances of 100 nanometers (10−9meters) or less or at very low temperature.
revolutionary
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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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