19.3 Electrical potential due to a point charge

 Page 1 / 2
• 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 $q$ from a large distance away to a distance of $r$ from a point charge $Q$ , and noting the connection between work and potential $\left(W=\phantom{\rule{0.25em}{0ex}}–q\Delta V\right)$ , it can be shown that the electric potential $V$ of a point charge is

$V=\frac{\text{kQ}}{r}\phantom{\rule{0.25em}{0ex}}\left(\text{Point Charge}\right),$

where k is a constant equal to $9.0×{\text{10}}^{\text{9}}\phantom{\rule{0.25em}{0ex}}\text{N}\phantom{\rule{0.25em}{0ex}}\text{·}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{\text{2}}\text{/}{\text{C}}^{\text{2}}$ .

Electric potential $V$ Of a point charge

The electric potential $V$ of a point charge is given by

$V=\frac{\text{kQ}}{r}\phantom{\rule{0.25em}{0ex}}\left(\text{Point Charge}\right).$

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

$\text{E}=\frac{\text{F}}{q}=\frac{\text{kQ}}{{r}^{2}}.$

Recall that the electric potential $V$ is a scalar and has no direction, whereas the electric field $\mathbf{\text{E}}$ 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 $V$ is closely associated with energy, a scalar, whereas $\mathbf{\text{E}}$ 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 $\left(\text{nC}\right)$ to microcoulomb $\left(\text{µC}\right)$ range. What is the voltage 5.00 cm away from the center of a 1-cm diameter metal sphere that has a $-3.00\phantom{\rule{0.25em}{0ex}}\text{nC}$ 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 $V=\text{kQ}/r$ .

Solution

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

$\begin{array}{lll}V& =& k\frac{Q}{r}\\ & =& \left(\text{8.99}×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{N}·{\text{m}}^{2}/{\text{C}}^{2}\right)\left(\frac{\text{–3.00}×{\text{10}}^{–9}\phantom{\rule{0.25em}{0ex}}\text{C}}{\text{5.00}×{\text{10}}^{\text{–2}}\phantom{\rule{0.25em}{0ex}}\text{m}}\right)\\ & =& \text{–539 V.}\end{array}$

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.

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.)

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

$V=\frac{\text{kQ}}{r}.$

Solution

Solving for $Q$ and entering known values gives

$\begin{array}{lll}Q& =& \frac{\text{rV}}{k}\\ & =& \frac{\left(0\text{.}\text{125}\phantom{\rule{0.25em}{0ex}}\text{m}\right)\left(\text{100}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{V}\right)}{8.99×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{N}·{\text{m}}^{2}/{\text{C}}^{2}}\\ & =& \text{1.39}×{\text{10}}^{–6}\phantom{\rule{0.25em}{0ex}}\text{C}=\text{1.39 µC.}\end{array}$

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.

is there more then 4 dimensions
hii
princy
hi
Miguel
hello I kinda need help in physics... a lot
Brown
Brown. what kind of help
Jeff
when it comes to physics stick with the basics don't overthink things
Jeff
yes
ayesha
sticking to the basics will take you farther than overwhelming yourself with more than you need to physics is simple keep it simple
Jeff
thk u Ayesha
Jeff
for real....? so I've got to know the fundamentals and use the formula to solve any problem
Brown
read Stephan hawkings a brief history of time
ayesha
ayesha
physics isn't hard it's just understanding and applying the formulas if u need help ask any question
ayesha
okay...because I've got an exam next year February a Computer based exam
Brown
ayesha
ayesha
how can we find absolute uncertainty
it what?
Luke
in physics
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
Luke
I have this one question can you please look it up it's 9702/22/O/N/17 Question 1 B 3
ayesha
what
uma
would you like physics?
Suthar
yes
farooq
precision or absolute uncertainty is always equal to least count of that instrument
Iram
how do I unlock the MCQ and the Essay?
what is the dimension of strain
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!
hi
Agboro
hiii
Chandan
Hi
Sahim
hi
Jeff
hey
Priscilla
sup guys
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
what s i unit of mass
Maxamed
SI unit of mass is Kg(kilogram).
Robel
what is formula of distance
Maxamed
Formula for for the falling body with initial velocity is:v^2=v(initial)^2+2*g*h
Mateo
i can't understand
Maxamed
we can't do this calculation without knowing the height of the initial position of the particle
Chathu
sorry but no more in science
Imoreh
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
50 N... (50 *1.732)N
Sahim
Plz cheak the ans and give reply..
Sahim
50 N...(50 *1.732)N
Ibrahim
show the working
usiomon
what is the value of f1 and f2
Syed
what is the value of force 1 and force 2.
Syed
.
Is earth is an inertial frame?
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
thanks
Irungu
Most welcome
Sahim
Hey.. I've a question.
Is earth inertia frame?
Sahim
only the center
Shii
What is an abucus?
Irungu
what would be the correct interrogation "what is time?" or "how much has your watch ticked?"
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
Young's modulus = stress/strain strain = extension/length (x/l) stress = force/area (F/A) stress/strain is F l/A x
El
so solve it
Ebenezer
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
what is cesium atoms?
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
I want learn physics
sir how to understanding clearly
Vinodhini
try to imagine everything you study in 3d
revolutionary
pls give me one title
Vinodhini
displacement acceleration how understand
Vinodhini
vernier caliper usage practically
Vinodhini
karthik sir is there
Vinodhini
what are the solution to all the exercise..?
What is realm
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
How to understand physics
i like physics very much
Vinodhini
i want know physics practically where used in daily life
Vinodhini
I want to teach physics very interesting to studentd
Vinodhini
how can you build interest in physics
Prince
Austin
understanding difficult
Vinodhini
vinodhini mam, physics is used in our day to day life in all events..... everything happening around us can be explained in the base of physics..... saying simple stories happening in our daily life and relating it to physics and questioning students about how or why its happening like that can make
revolutionary
revolutionary
anything send about physics daily life
Vinodhini
How to understand easily
Vinodhini
revolutionary
even when you see this message in your phone...it works accord to a physics principle. you touch screen works based on physics, your internet works based on physics, etc....... check out google and search for it
revolutionary
why is it dat when using double pan balance the known and unknown mass are the same