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McMaster University

Physics

PHYSICS 1E03

Waldemar Okon

Fall

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Part 1: Mechanics
Chapter 1: Physics and Measurement
The three fundamental physical quantities of mechanics are length, mass, and time, which
in the SI system have the units metres (m), kilograms (kg), and seconds (s), respectively.
Prefixes indicating various powers of ten are used with these three basic units. The
density of a substance is defined as its mass per unit volume. Different substances have
different densities mainly because of differences in their atomic masses and atomic
arrangements.
The number of particles in one mole of any element or compound, called Avogadro’s
23
number, N ,Ais 6.02·10 .
The method of dimensional analysis (also called the factor label method) is a very
powerful tool in solving physics problems. Dimensions can be treated as algebraic
quantities. By making estimates and making order-of-magnitude calculations, you should
be able to approximate the answer to a problem when there is not enough information
available to completely specify an exact solution.
When you compute a result from several measured numbers, each of which has a certain
accuracy, you should give the result with the correct number of significant figures.
SI Prefixes
Power Prefix Abbreviation Power Prefix Abbreviation
101 deka da 10 -1 deci d
2 -2
10 hecta h 10 centi c
103 kilo k 10 -3 milli m
6 -6 m
109 mega M 10 -9 micro
10 giga G 10 nano n
1012 tera T 10 -12 pico p
15 -15
10 peta P 10 femto f
1018 exa E 10 -18 atto a
1021 zetta Z 10 -21 zepto z
24 -24
10 yotta Y 10 yocto y
Chapter 2: Motion in One Dimension
After a particle moves along the x axis from some initial position x to some final
i
position xf, its displacement is
Dx ” xf- x i The average velocity of a particle during some time interval is the displacement Dx
divided by the time interval Dt during which that displacement occurred:
Dx
v x
Dt
The average speed of a particle is equal to the ratio of the total distance it travels to the
total time it takes to travel that distance.
The instantaneous velocity of a particle is defined as the limit of the ratio Dx/ Dt as Dt
approaches zero. By definition, this limit equals the derivative of x with respect to t, or
the time rate of change of the position:
Dx dx
vx” lDtﬁ0 ”
Dt dt
The instantaneous speed of a particle is equal to the magnitude of its instantaneous
velocity.
The average acceleration of a particle is defined as the ratio of the change in its velocity
Dv dxvided by the time interval Dt during which that change occurred:
Dv x vxfv xi
ax” =
Dt tf-t i
The instantaneous acceleration is equal to the limit of the ratio Dv /Dt as Dt
x
approaches zero. By definition, this limit equals the derivative of v with respect to t, or
x
the time rate of change of the velocity:
2
Dv x d d
ax” lim = vx= 2 x
D ﬁ0 Dt dt dt
The equations of kinematics for a particle moving along the x axis with uniform
acceleration a (cxnstant in magnitude and direction) are
vxf v +xit x
xf- x i v tx= 2 (vxiv xf)t
x - x = v t + a t1 2
f i xi 2 x
2 2
vxf = vxi + 2a x (xf- x i)
You should be able to use these equations and the definitions in this chapter to analyze
the motion of any object moving with constant acceleration.
An object falling freely in the presence of the Earth’s gravity experiences a free-fall
acceleration directed toward the centre of the Earth. If air resistance is neglected, if the
motion occurs near the surface of the Earth, and if the range of the motion is small compared with the Earth’s radius, then the free-fal2 acceleration g is constant over the
range of motion, where g is equal to 9.80m/s .
Complicated problems are best approached in an organized manner. GOAL Problem
Solving involves four steps: Gather information, Organize your approach, Analyze the
problem, and Learn from your efforts.
Chapter 3: Vectors
Scalar quantities are those that have only magnitude and no associated direction. Vector
quantities have both magnitude and direction and obey the laws of vector addition.
We can add two vectors A and B graphically, using either the triangle method or the
parallelogram rule. In the triangle method the resultant vector R = A +B runs from the
tail of A to the tip of B. In the parallelogram method R is the diagonal of a parallelogram
having A and B as two of its sides.
The x component A of the vector A is equal to the projection of A along the x axis of a
x
coordinate system, as shown below, where A = Acosq .xThe y component A of A is y
the projection of A along the y axis, where A = Asiny .
If a vector A has an x component A and a y component A , the vector can be expressed
x y
in unit-vector form as A = A i + A j. In this notation, i is a unit vector pointing in the
x y
positive x direction, and j is a unit vector pointing in the positive y direction. Because i
and j are unit vectors, i = j =1 . We can find the resultant of two or more vectors by resolving all vectors into their x and y
components, adding their resultant x and y components, and then using the Pythagorean
theorem to find the magnitude of the resultant vector. We can find the angle that the
resultant vector makes with respect to the x axis by using a suitable trigonometric
function.
Chapter 4: Motion in Two Dimensions
If a particle moves with constant acceleration a and has velocity v and position r ati i
t = 0, its velocity and position vectors at some later time t are
v = v +at
f i
1 2
rf= r +iv t +iat 2
For two-dimensional motion in the xy plane under constant acceleration, each of these
vector expressions is equivalent to two component expressions – one for the motion in
the x direction and one for the motion in the y direction.
Projectile motion is one type of two-dimensional motion under constant acceleration,
where a = 0xand a = -g . Iy is useful to think of projectile motion as the superposition
of two motions:
1. Constant-velocity motion in the x direction: v = v = v cosq xfd xi i i
2. Free-fall motion in the vertical direction subject to a constant downward acceleration
of magnitude g = 9.80 m/s : 2 v = v -9.80 t m (assuming up is positive)
yf yi s2
where q is ihe angle that the initial velocity vector v makes with tie x axis.
A particle moving in a circle of radius r with a constant speed v is in uniform circular
motion. It undergoes a centripetal (or radial) acceleration a because the dirrction of v
changes in time. The magnitude of a is r
v 2
ar=
r
and its direction is always toward the centre of the circle.
If a particle moves along a curved path in such a way that both the magnitude and the
direction of v change in time, then the particle has an acceleration vector that can be
described by two component vectors:
1. a radial component vector a that causer the change in direction of v and
2. a tangential component vector a that causestthe change in magnitude of v.
2
The magnitude of a is v /rr, and the magnitude of a is t d v /dt . The velocity v of a particle measured in a fixed frame of reference S can be related to the
velocity v'of the same particle measured in a moving frame of reference S 'by
v' = v - v0
where v is the velocity of S 'relative to S.
0
Chapter 5: The Laws of Motion
Newton’s Laws of Motion
Newton' s First Law: In the absence of an external force, a body at rest remains at
rest and a body in uniform motion in a straight line maintains that motion.
▯ An inertial frame is one that is not accelerating.
Newton' s Second Law: The acceleration of an object is directly proportional to the net
force acting on it and inversely proportional to its mass.
▯ The net force acting on an object equals the product of its mass and its
acceleration: ▯ F = ma .
▯ The same relationship holds for the components of force and acceleration in a
specific direction, such as x or y▯ Fx= ma x, ▯ F y ma y
▯ If an object is either stationary or moving with constant velocity(there is no
net acceleration), then the forces must vectorially cancel each other out(there
is no net force): 0 net acceleration = 0 net force.
Newton' s Third Law: If two objects interact, then the force exerted by object 1 on
object 2 is equal in magnitude and opposite in direction to the force exerted by object
2 on object 1.
▯ An isolated force cannot exist in nature.
▯ Application to Star Wars: Without the dark side there can be no light side.
Gravity (Near the Earth’s Surface)
The force of gravity exerted on an object is equal to the product of its mass (a scalar
quantity) and the free-fall acceleration: F = mg . The weight of an object is the
g
magnitude of the force of gravity acting on the object.
Friction
The maximum force of static friction f between an object and a surface is
smax
proportional to the normal force acting on the object. In general,
f £ m n
s s
where f rspresents the magnitude of the maximum force of static friction, m is she
coefficient of static friction and n is the magnitude of the normal force.
When an object slides over a surface, the direction of the force of kinetic friction f is
k
opposite the direction of sliding motion and is also proportional to the magnitude of the
normal force. The magnitude of this force is given by k = m k
where m ik the coefficient of kinetic friction.
Chapter 6: Circular Motion and Other Applications of Newton’s Laws
Centripetal Acceleration
A radial force is one that is directed perpendicular to the direction of motion of the
object on which the force acts. Newton’s second law applied to a particle moving in
uniform circular motion states that the net force causing the particle to undergo a
centripetal acceleration is
mv 2
▯ Fr= ma =r
r
that is, the sum of the forces in the radial direction(radial forces) is equal to the particle’s
mass times the radial acceleration, which is equal to the particle’s mass times the
magnitude of the velocity squared divided by the radius of the circle being moved in.
A particle in nonuniform circular motion has both a centripetal component of acceleration
and a nonzero tangential component of acceleration. In the case of a particle rotating in a
vertical circle, the force of gravity provides the tangential component of acceleration and
part or all of the centripetal component of acceleration.
Applying Newton’s Laws from a Noninertial Reference Frame
An observer in a noninertial (accelerating) frame of reference must introduce fictitious
forces when applying Newton’s second law in that frame. If these fictitious forces are
properly defined, the description of motion in the noninertial frame is equivalent to that
made by an observer in an inertial frame. However, the observers in the two frames do
not agree on the causes of the motion.
Resistive Forces
A body moving through a liquid or gas experiences a resistive force that is speed-
dependent. This resistive force, which opposes the motion, generally increases with
speed. The magnitude of the resistive force depends on the shape of the body and on the
properties of the medium through which the body is moving. In the limiting case for a
falling body, when the magnitude of the resistive force equals the body’s weight, the
body reaches its terminal speed.
Chapter 7: Work and Kinetic Energy
Work –Constant Force
The work done by a constant force F acting on a particle is defined as the product of the
component of the force in the direction of the particle’s displacement and the magnitude
of the displacement. Given a force F that makes an angle q with the displacement vector d of a particle acted on by the forced, you should be able to determine the work done by
F using the equation
W ” Fd cosq
The unit for work is the Joule (J), defined as the product of a Newton and a metre:
J = N×m
Scalar Product
The scalar product (A.K.A. dot product) of two vectors A and B is defined by the
relationship
A×B ” ABcosq
where the result is a scalar quantity and q is the angle between the two vectors. The
scalar product obeys the commutative and distributive laws.
Work –Varying Force
If a varying force does work on a particle as the particle moves along the x axis from x
i
to x , you must use the expression
f
xf
W ” ▯x F xx
i
where F ix the component of force in the x direction. If several forces are acting on the
particle, the net work done by all of the forces is the sum of the amounts of work done by
all of the forces.
Kinetic Energy
The kinetic energy of a particle of mass m moving with a speed v (where v is small
compared with the speed of light) is
2
K ” m2
The Work-Kinetic Energy Theorem
The work-kinetic energy theorem states that the net work done on a particle by external
forces equals the change in kinetic energy of the particle:
2 2
▯ W = K -Kf= mvi- m2 f 2 i
If a frictional force acts, then the work-kinetic energy theorem can be modified to give
K + W - f d = K
i ▯ other k f
Power
The instantaneous power P is defined as the time rate of energy transfer. If an agent
applies a force F to an object moving with a velocity v, the power delivered by that agent
is
dW
P ” = F×v
dt Chapter 8: Potential Energy and Conservation of Energy
Gravitational and Elastic Potential Energy
If a particle of mass m is at a distance y above the Earth’s surface, the gravitational
potential energy of the particle-Earth system is
U = mgy
g
The elastic potential energy stored in a spring of force constant k displaced a distance x
from its equilibrium position is
U ” kx 1 2
s 2
Conservative and Nonconservative Forces
A force is conservative if:
1. the work it does on a particle moving between two points is independent of the path
the particle takes between the two points.
2. the work it does on a particle is zero when the particle moves through an arbitrary
closed path and returns to its initial position.
▯ Gravity is a conservative force.
A force that does not meet these criteria is said to be nonconservative.
▯ Friction is a nonconservative force.
Potential Energy Functions
A potential energy function U can be associated only with a conservative force. If a
conservative force F acts on a particle that moves along the x axis from x to x , then the
i f
change in the potential energy of the system equals the negative of the work done by that
force:
xf
U fU = -i ▯ F xx
i
Total Mechanical Energy
The total mechanical energy of a system is defined as the sum of the kinetic energy and
the potential energy:
E ” K +U
If no external forces do work on a system and if no nonconservative forces are acting on
objects inside the system, then the total mechanical energy of the system is constant:
K iU = i +U f f
If nonconservative forces (such as friction) act on objects inside a system, then the
mechanical energy is not conserved. In these situations, the difference between the total
final mechanical energy and the total initial mechanical energy of the system equals the
energy transferred to or from the system by the nonconservative forces.
Chapter 9: Linear Momentum and Collisions Linear Momentum
The linear momentum p of a particle of mass m moving with a velocity v is
p ” mv
Linear Momentum is Conserved
The law of conservation of linear momentum indicates that the total momentum of an
isolated system is conserved. If two particles form an isolated system, their total
momentum is conserved regardless of the nature of the force between them. Therefore,
the total momentum of the system at all times equals its initial total momentum, or
p1ip =2i 1f+p 2f
The Impulse-Momentum Theorem
The impulse imparted to a particle by a force F is equal to the change in the momentum
of the particle:
f
I ”▯tiFdt = Dp
This is known as the impulse-momentum theorem.
• Impulsive forces are often very strong compared with other forces on the system and
usually act for a very short time, as in the case of collisions.
Elasticity of Collisions
When two particles collide, the total momentum of the system before the collision always
equals the total momentum after the collision, regardless of the nature of the collision.
• An inelastic collision is one for which the total kinetic energy is not conserved.
• A perfectly inelastic collision is one in which the colliding bodies stick together after
the collision.
• An elastic collision is one in which kinetic energy is conserved.
• In a two- or three-dimensional collision, the components of momentum in each of the
three directions (x, y, and z) are conserved independently.
Centre of Mass
The position vector of the centre of mass of a system of particles is defined as
mr
▯ i i
rCM = i
M
where M = m is the total mass of the system and r is the position vector of the ith
▯ i i
i
particle.
• The position vector of the centre of mass of a rigid body can be obtained from the
integral expression
1
rCM = ▯ rdm
M
• The velocity of the centre of mass of a system of particles is ▯ m i i
vCM = i
M
• The total momentum of a system of particles equals the total mass multiplied by the
velocity of the centre of mass.
p = m v
system▯i i i
Newton’s Second Law Applied to a System of Particles
Newton’s second law applied to a system of particles is
dp
▯ Fext Ma CM = tot
dt
where a is the acceleration of the centre of mass and the sum is over all external
CM
forces. The centre of mass moves like an imaginary particle of mass M under the
influence of the resultant external force on the system. It follows that the total
momentum of the system is conserved if there are no external forces acting on it.
Chapter 10: Rotation of a Rigid Object About a Fixed Axis
Angular Mechanics
• If a particle rotates in a circle of radius r through an angle q (measured in radians),
the arc length it moves through is s = rq .
• The angular displacement of a particle rotating in a circle or of a rigid object
rotating about a fixed axis is
Dq =q -f i
• The instantaneous angular speed of a particle rotating in a circle or of a rigid object
rotating about a fixed axis is
w = dq
dt
• The instantaneous angular acceleration of a rotating object is
dw d q
a = = 2
dt dt
• When a rigid object rotates about a fixed axis, every part of the object has the same
angular speed and the same angular acceleration.
Equations of Angular Kinematics
If a particle or object rotates about a fixed axis under constant angular acceleration, one
can apply the angular equations of kinematics that are analogous to those for linear
motion under constant linear acceleration:
• w =w +at
f i
• q =q +wt + at 1 2
f i i 2 • w =w +2a q -q
f i ( f i)
• A useful technique in solving problems dealing with rotation is to visualize a linear
version of the same problem.
Converting from Linear to Angular
When a rigid object rotates about a fixed axis, the angular position, angular speed, and
angular acceleration are related to the linear position, linear speed, and linear acceleration
through the relationships:
• s = rq
• v = rw
• at= ra
Moment of Inertia of a System of Particles
The moment of inertia of a system of particles is
I ” m r 2
▯i i i
Rotational Energy
If a rigid object rotates about a fixed axis with angular speew , its rotational energy
can be written
K = Iw1 2
R 2
where I is the moment of inertia about the axis of rotation.
Moment of Inertia of a Rigid Body
The moment of inertia of a rigid object is
2
I = ▯ dm
where r is the distance from the mass element dm to the axis of rotation.
Torque
The magnitude of the torque associated with a force F acting on an object is
t = Fd
where d is the moment arm of the force, which is the perpendicular distance from some
origin to the line of action of the force. Torque is a measure of the tendency of the force
to change the rotation of the object about some axis.
If a rigid object free to rotate about a fixed axis has a net external torque acting on it,
the object undergoes an angular acceleration a , where
▯ t = Ia Power Delivered by an External Force in Rotating a Rigid Object About a Fixed
Axis
The rate at which work is done by an external force in rotating a rigid object about a
fixed, or the power delivered, is
P =tw
The Rotational-Work-Kinetic Energy Theorem
The net work done by external forces in rotating a rigid object about a fixed axis equals
the change in the rotational kinetic energy of the object:
W = Iw - Iw 1 2
▯ 2 f 2 i
Chapter 16: Wave Motion
A transverse wave is one in which the particles of the medium move in a direction
perpendicular to the direction of the wave velocity. An example is a wave on a taut
string. A longitudinal wave is one in which the particles of the medium move in a
direction parallel to the direction of the wave velocity. Sound waves in fluids are
longitudinal. You should be able to identify both types of waves.
Any one-dimensional wave traveling with a speed of v in the x direction can be
represented by a wave function of the form
y = f(x–vt )
where the positive sign applies to a wave traveling in the negative x direction and the
negative sign applies to a wave traveling in the positive x direction. The shape of the
wave at any instant in time (a snapshot of the wave) is obtained by holding t constant.
The superposition principle specifies that when two or more waves move through a
medium, the resultant wave function equals the algebraic sum of the individual wave
functions. When two waves combine in space, they interfere to produce a resultant wave.
The interference may be constructive (when the individual displacements are in the
same direction) or destructive (when the displacements are in opposite directions).
The speed of a wave traveling on a taut string of mass per unit length m and tension T is
T
v = m
A wave is totally or partially reflected when it reaches the end of the medium in which it
propagates or when it reaches a boundary where its speed changes discontinuously. If a wave pulse traveling on a string meets a fixed end, the pulse is reflected and inverted. If
the pulse reaches a free end, it is reflected but not inverted.
The wave function for a one-dimensional sinusoidal wave traveling to the right can be
expressed as
2p
y= Asin ▯ (x-vt ▯ = Asin(kx -t )
▯l ▯
where A is the amplitude, l is the wavelength, k is the angular wave number,wand
is the angular frequency. If T is the period and f the frequency, v, k and w can be
written
v = l = l f
T
k ” 2p
l
2p
w ” =2p f
T
You should know how to find the equations describing the motion of particles in a wave
from a given set of physical parameters.
Chapter 18: Superposition and Standing Waves
When two traveling waves having equal amplitudes and frequencies superimpose, the
resultant wave has an amplitude that depends on the phase angle f between the two
waves. Constructive interference occurs when the two waves are in phase,
corresponding to f = 0,2p,4p,...rad. Destructive interference occurs when the two
o
waves are 180 out of phase, corresponding to f =p,3p,5p...rad. Given two wave
functions, you should be able to determine which if either of these two situations applies.
Standing waves are formed from the superposition of two sinusoidal waves having the
same frequency, amplitude, and wavelength but traveling in opposite directions. The
resultant standing wave is described by the wave function
y =(2Asinkx )oswt
Hence, the amplitude of the standing wave is 2A, and the amplitude of the simple
harmonic motion of any particle of the medium varies according to its position as
2Asinkx. The points of zero amplitude (called nodes) occur at x = nl /2 (n = 0, 1, 2, 3,
...). The maximum amplitude points (called antinodes) occur at x = nl /4 (n = 1, 3, 5,
...). Adjacent antinodes are separated by a distance l /2. Adjacent nodes also are
separated by a distance l /2. Nodes and antinodes that are adjacent to each other are
separated by a distance l /4.
The natural frequencies of vibration of a taut string of length L and fixed at both ends are n T
n = 2L m n =1,2,3,..
where T is the tension in the string and m is its linear mass density. The natural
frequencies of vibration f 12 f1,3f1,... form a harmonic series.
An oscillating system is in resonance with some driving force whenever the frequency
of the driving force matches one of the natural frequencies of the system. When the
system is resonating, it responds by oscillating with a relatively large amplitude.
Standing waves can be produced in a column of air inside a pipe. If the pipe is open at
both ends, all harmonics are present and the natural frequencies of oscillation are
v
fn= n 2L n =1,2,3,...
If the pipe is open at one end and closed at the other, only the odd harmonics are present,
and the natural frequencies of oscillation are
f = n v n =1,3,5,...
n 4L
The phenomenon of beating is the periodic variation in intensity at a given point due to
the superposition of two waves having slightly different frequencies.
Chapters:
23-27
Chapter 23: Electric Fields
Electric charges have the following important properties:
• Unlike charges attract one another, and like charges repel one another.
• Charge is conserved.
• Charge is quantized—that is, it exists in discrete packets that are some integral
multiple of the electronic charge.
Conductors are materials in which charges move freely. Insulators are materials in
which charges do not move freely.
Coulomb’s law states that the electric force exerted by a charge q on a second
1
charge q is
2
q1 2
F12 k e 2 r
r where r is the distance between the two charges and r is a unit vector directed from q to 1
9 2
8.99·10 N×m
q 2 The constant k , ealled the Coulomb constant, has the value k = e C 2 .
The smallest unit of charge known to exist in nature is the charge on an electron
or proton, e =1.602 19·10 C -19 .
The electric field E at some point in space is defined as the electric force F thae
acts on a small positive test charge placed at that point divided by the magnitude of the
test charge q 0
F e
E =
q0
At a distance r from a point charge q, the electric field due to the charge is given
by
q
E = k e 2rˆ
r
where r is a unit vector directed from the charge to the point in question. The electric
field is directed radially outward from a positive charge and radially inward toward a
negative charge.
The electric field due to a group of point charges can be obtained by using the
superposition principle. That is, the total electric field at some point equals the vector
sum of the electric fields of all the charges:
q
E = k e▯ i ri
i ri
The electric field at some point of a continuous charge distribution is
dq
E = k e▯ 2rˆ
r
where dq is the charge on one element of the charge distribution and r is the distance
from the element to the point in question.
Electric field lines describe an electric field in any region of space. The number
of lines per unit area through a surface perpendicular to the lines is proportional to the
magnitude of E in that region.
A charged particle of mass m and charge q moving in an electric field E has an
acceleration
qE
a =
m Chapter 24: Gauss’s Law
Electric flux is proportional to the number of electric field lines that penetrate a surface.
If the electric field is uniform and makes an angleq with the normal to a surface of area
A, the electric flux through the surface is
F E EAcosq
In general, the electric flux through a surface is
F E ▯ E×dA
surface
These equations are most useful when symmetry simplifies the calculation.
Gauss’s law says that the net electric fluxF through any closed gaussian surface
E
is equal to the net charge inside the surface divide0 by ˛ :
q in
F E ▯ E×dA = ˛
0
Charge Distribution Electric Location
Field
Q
ke
Insulating sphere of radius R, uniform charge
r2 r > R
density, and total charge Q ▯ Q r < R
k r
▯ eR 3
Q r > R
Thin spherical shell of radius R and total chargeker 2
Q ▯ r < R
▯ 0
Line charge of infinite length and charge per un2k l Outside the line
length l e r
Nonconducting, infinite charged plane having s Everywhere outside
surface charge densits 2˛ the plane
0
s
Just outside conductor
Conductor having surface charge densits ▯˛0
0 Just inside conductor
▯
Chapter 25: Electric Potential
When a positive test charge0q is moved between points A and B in an electric field E, the
change in potential energy is B
DU = -q 0 ▯AE×ds
U
The electric potential, V = is a scalar quantity and has units of joules per
q 0
coulomb(J/C), where 1 J/C = 1 V.
The potential difference DV between points A and B in an electric field E is
defined as
DU B
DV = = - E×ds
q 0 ▯A
The potential difference between two points A and B in a uniform electric field E
is
DV = -Ed
where d is the magnitude of the displacement in the direction parallel to E.
An equipotential surface is one on which all points are at the same electric
potential. Equipotential surfaces are perpendicular to electric field lines.
If we define V = 0 at rA= ¥ , the electric potential due to a point charge any
distance r from the charge is
q
V = k er
We can obtain the electric potential associated with a group of point charges by summing
the potentials due to the individual charges.
The potential energy associated with a pair of point charges separated by a
distance r12s
q q
U = ke 1 2
12
This energy represents the work required to bring the charges from an infinite separation
to the separation

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