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Electric Field Content 1 Concept of an electric field 2 Force between point charges 3 Electric field of a point charge 4 Uniform electric fields 5 Electric potential Force between point charges Coulomb’s law states that the magnitude of the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of their distance apart.
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Coulomb’s law can be expressed mathematically as:
F
1 Q1Q2 4 0 r 2
where Q1 and Q2 are two point charges which are situated a distance r apart in a medium; where o is the permittivity of free space or vacuum and o = 8.85 x 10-12 farad per metre ( F m-1 )
Coulomb’s law is analogous to Newton’s law of gravitation. Like charges repel and unlike charges attract.
Concept of an electric field An electric field is a region in which an electric charge experiences a force.
The direction of the field at any point is defined as the direction of the force experienced by a small, positive charge placed at that point.
For example, the positive charge placed between the plates will experience a force in the direction shown. As in gravitation and magnetism, we make use of field lines to represent electric fields. The direction of the arrows indicates the direction of the field at any point.
Electric field lines are more dense the larger the magnitude of electric field.
When the lines are parallel, the field is uniform (which means strength and direction are constant), as shown in the figure below, the lines are parallel and equally-spaced near the centre of the region between two parallel plates.
When lines are spreading out, the field pattern is radial. For a positive charge, the direction of the electric field is radially outwards. For a negative charge, the direction of the electric field is radially inwards. The electric field decreases as distance from the charge is increased, as the number of lines per unit area decreases.
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Rules for drawing Electric field lines Electric field lines: 1. Point in the direction of the electric field. 2. Start at positive charge (+). 3. End at negative charge (-). 4. are more dense where electric field has a larger magnitude. Examples of non uniform fields
Electric field lines between two like charges
Electric field lines between unlike charges
Electric field lines between a +2q and a –q charges Electric Field Strength The electric field strength at a point in an electric field is the force per unit charge exerted on a test charge placed at that point. Mathematically, the electric field strength at point P is given by:
E
From
E
F q
-1
units of E = N C (newton per coulomb)
F , the force exerted on a charge q at a point where the electric field strength is E is given by q
F qE Since F is a vector and
E
F , it should be noted that E is also a vector. q
Electric field strength E = F / q is analogous to gravitational field strength g = F / m.
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Electric field of a point charge: radial electric field Consider a small test charge (+q) placed in air at a point A, which is distance (r) from a point charge (+Q).
The force acting on the positive test charge +q is radially outwards from +Q From Coulomb’s law, the force (F) acting on the small positive charge +q due to the charge (+Q) is given by:
F
1 Qq 4 0 r 2
From the definition of electric field strength, the field strength E at point A is given by:
E
E
F q
1 Q 4 0 r 2
The electric field strength around the charge is inversely proportional to the square of the distance from the charge. The electric field follows the inverse square law. For a positive charge +Q, the direction of E is radially outwards from the centre of +Q. For a negative charge -Q, the direction of E is radially inwards towards the centre of -Q. Since E is a vector, when calculating the resultant electric field at a point due to a number of charges, the electric fields must be added vectorially. Electric field strength E is analogous to gravitational field strength g.
Uniform electric fields The figure below shows the uniform electric field of strength (E) between two metal plates (A and B) which are separated by a distance (d) and have a potential difference (V) between them.
Electric field strength in a uniform electric field between two parallel plates, with separation d, is given by:
E
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V d
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The units of electric field is newton per coulomb (N C ) or volt per meter (V m ). The equation above applies only if the electric field is uniform (i.e. constant magnitude and same direction at all points).
Electric Potential The electric potential (V) at a point in an electric field is defined as the work done per unit positive charge by an external agent from infinity to that point. Mathematically, V
W q
where W is the work done to bring the charge q from infinity to the point Note : The units of potential is the volt or joule per coulomb. 1 volt is defined as 1 joule per coulomb From the definition of potential, the zero electric potential is at an infinite distance from a charge. However, for most practical purposes, we define the zero of electric potential to be that of the Earth. This is because the potential of the Earth is practically constant. Electric potential is a scalar quantity. When more than one field is present, the net potential is the algebraic sum of the potentials due to each field. However, electric field strength is a vector quantity and calculation of resultant electric field strength involves vector addition of individual electric field strengths. Electric potential due to a point charge The figure shows an isolated point charge (+Q) situated at a point A At B, distance r from the point charge, the electric potential V is given by:
V
1 Q 4 0 r
At an infinite distance from the point charge ( r = ), the electric potential is zero. Electric potential decreases with distance r from an isolated point charge. Points which are equal distances from the center of a radial field, in any direction, have equal potential. The points on a circle with its centre at A have the same potential. The potential at a point due to a number of point charges is equal to the algebraic sum of the potentials at that point due to each of the charges.
Equipotential Surfaces An equipotential surface is the surface over which the potential in an electric field remains constant. No energy change occurs and no work is done when a charge moves along such a surface. For the field due to an isolated point charge, the equipotential surfaces are concentric spherical surfaces.
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At any point, the direction of the electric field is perpendicular to the equipotential surface.
Electric Potential Energy Electric potential energy (U) of a charge q in an electric field where the electric potential is V, is given by U=qV The electric potential energy U of a test charge q at a distance r from a point charge Q is given by:
W
1 Qq 4 0 r
Electric Potential Difference In practice, we are usually concerned with potential difference (p.d.) between two points rather than with absolute values of potential. Definition of potential difference
The potential difference between two points in an electric field is the work done (or energy changed) per unit charge moving from one point to the other. i.e. potential difference, V
W q
where W is the work done in moving the charge from one point to the other. The units of potential difference is the volt or the joule per coulomb 1 volt of potential difference is defined as 1 joule per unit coulomb of work done to move the charge from one point to the other. Consider an electron moving in a vacuum between two points which have a p.d. of 1 volt between them. The electron accelerates under the influence of electric force. Energy gained by the electron = work done by the electric force = qV -19
W = (1.610 C)(1 V) -19 = 1.610 J
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-19
the charge on an electron = 1.610
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The interaction between an incoming alpha particle and a stationary nucleus An electric force acts on a charge situated in an electric field. If the charge is moved over a distance against this force, work is done against the electric force and the system gains electrical potential energy. The amount of potential energy gained is equal to the work done in moving the charge. Figure below shows the behavior of an alpha-particle, which approximates to a point charge, in the electric field surrounding a large nucleus. An alpha-particle, being positively charged, experiences a repulsive force everywhere in the field due to the nucleus. If the alpha-particle approaches the head-on, the repulsive force causes the alpha-particle to lose kinetic energy, and the system gains an equal amount of electrical potential energy. The alpha-particle will be brought to rest when the gain of electrical potential energy of the system is equal to the initial kinetic energy of the particle. The particle will then reverse its direction and be accelerated away from the nucleus, as the system returns its electrical potential energy and gains kinetic energy. The repulsive force is felt by both bodies, but, because the alpha-particle has much less mass than the nucleus, it will gain the vast majority of the kinetic energy.
Electric Potential Gradient
In general, the field strength of the field at a point is related to the potential by
E
dV . dx
dV is the potential gradient i.e. the change in potential per unit distance. dx Hence, electric field strength is the negative potential gradient at that point. For uniform field dV / dx = V / d
where V is the potential difference and d is the distance
Hence E = V / d Electric field at a point is numerical equal to V / d
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Example In discussing electric fields, the terms ‘electric field strength’, ‘electric potential’ and ‘potential gradient’ are used. Which statement about these terms is correct? A Electric field strength at a point is the work done in bringing unit positive charge from infinity to the point. B Electric potential and potential gradient are both scalar quantities. C The potential gradient at a point is numerically equal to the electric field strength at that point. D Unit potential gradient exists between any two points, if one joule of work is done in transporting one coulomb of charge between the points. (N99/I/Q16) Motion of Charged Particles in an Electric Field When a particle of charge q and mass m is placed in an electric field E, the electric force on the charge is qE. Then, the mass experiences a force and an acceleration given by
F qE ma
a
qE m
If the electric field is uniform (that is, constant in magnitude and direction), the acceleration is a constant. When the charged particle accelerates, it gains kinetic energy while losing electrical potential energy. kinetic energy gained = loss in electrical potential energy
1 mv 2 qV 2 where v is the speed of charged particles and V is the potential difference between the two parallel charged plates. If the charge is positive, the acceleration is in the direction of the electric field. If the charge is negative, the acceleration is in the direction opposite the electric field. Note that positive charged particle is attracted to negative plate and vice versa
Negatively charged plate
Positively charged plate
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If an electron is projected horizontally into a vertical uniform electric field as shown below, it experiences a constant force in a direction opposite to E, hence the path of motion of the electron is parabolic.
+++++++++++++ Example A charged particle is projected horizontally at P into a uniform vertical field. The particle follows the path shown.
Ignoring gravitational effects, what describes a possible state of charge of the particle and the nature of the field? Charge field A Negative electric B Negative magnetic C Positive electric D Positive magnetic (J2000/I/Q27) Example An electron is projected at right angle to a uniform electric field E.
In the absence of other fields, in which direction is the electron deflected? A into the plane of the paper B out of the plane of the paper C to the left D to the right. (N98/I/Q17)
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Analogy with Gravitational Fields Electric Field Force equation
Gravitational Field
Coulomb’s law
Newton’s law of gravitation
1 Q1Q 2 F 4 0 r 2
F G
Quantity which generates the field
Charge q
Mass m
Quantity which experiences the force Constant of proportionality
Charge Q
Mass M
1 4 0
G
Relationship of force with distance r Direction of force
Mm r2
The universal gravitational constant independent of the medium
where 0 is the permittivity of the medium, whose value depends on the nature of the medium. 2 Inversely proportional to r Inverse square law Like charges repel. Unlike charges attract.
Inversely proportional to r Inverse square law Always attractive.
Definition of field strength
E=F/q
Field due to point charge / mass
E
1 Q 4 0 r 2
g G
Potential due to point charge / mass
V
1 Q 4 0 r
G
2
g=F/m
M r2 M r
Natural phenomenon and Application of electric fields (Optional)
In stormy weather, the potential difference between Earth and the atmosphere can be so great that a sudden current discharge (lightning) takes place. Lightning is an awesome and spectacular example of electricity in action. Friction within clouds can cause clouds to become highly charged. Eventually the charge becomes so great that the insulation of the air breaks down, causing the electric charge to flow to the ground as an enormous spark. The energy of the discharge is so great that it produces an intense trail of light, heat and sound. The transmission of electrical impulses along a nerve in your body can be explained in terms of nd electric field. See Physics 2 Edition by Robert Hutchings Pg284 for details. An electric eel uses its electric field to detect the presence of its prey. Capacitors are manufactured electrical components, they are used to store energy in the electric field between charged conductors. Electrostatic crop spraying is an example of application of electric field. The nozzle of the spray is maintained at higher potential than the plant below. Droplets from the nozzle acquired positive charge, and thus will be attracted to the plant (which has negative charge). The spray will coat the whole plant, even under the leaves where many pests reside. Lightning rods on the rooftop allow charges from the clouds to be neutralised, so protect the building from being damaged by lightning. Smoke cleaning- ash particles in the smoke can be removed from the gas so that it will not cause pollution. Photocopiers uses electrostatics to copy an image.
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Summary Page on Electric Field
Coulomb’s Law: The force between two point charges in free space(vacuum) is
F
1 Q1Q 2 4 0 r 2
An electric field is a region in which an electric charge experiences a force.
Electric field strength(intensity) at a point is the force acting on a unit positive charge at that point.
Electric field strength is a vector quantity.
Radial electric field Electric field strength of a charged particle Q in free space is
E
1 Q 4 0 r 2
Uniform electric field Electric field strength between two charged parallel plates (separation d) is
E
V d
Electric potential at a point is defined as the work done per unit charge in bringing a positive unit charge from infinity to that point.
Electric potential at a point distance r from a point charge Q is given by
V
1 Q 4 0 r
Electric field is numerically equal to the potential gradient.
E
dV dx
The End
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