Unit 3: Electric Potential
Lesson 3.6
Calculating Electric Field
Contents
Introduction
1
Learning Objectives
2
Warm Up
2
Learn about It!
Coulomb’s Law and the Electric Field
Electric Field of a Single Point Charge
Electric Field at a Point between Two Charges
Two-Dimensional Electric Fields
3
4
5
5
6
Key Points
13
Key Formulas
13
Check Your Understanding
15
Challenge Yourself
17
Bibliography
18
Key to Try It!
18
Unit 3: Electric Potential
Lesson 3.6
Calculating Electric Field
Introduction
Contrary to popular belief, sharks are not attracted by blood in their pursuit for food. Sharks
possess a highly specialized organ called the “ampullae of Lorenzini,” which contains
electroreceptor cells that enable them to detect the electric fields produced by their prey.
These tube-jellies under their skin are considered by most as one of the most evolved
biological mechanisms known to man. In this lesson, we will see the mathematical wonders
that surround the concept of the electric field.
3.6. Calculating Electric Field
1
Unit 3: Electric Potential
Learning Objectives
In this lesson, you should be able to do the
following:
●
DepEd Competency
Calculate the electric field in the
region given a mathematical
function describing its potential in
Describe the magnitude and direction
of the electric field around a charged
a region of space
(STEM_GP12EMIIIc-20).
object.
●
Express the general formula in
determining electric field.
●
Calculate the electric field between
two charges in one and in two
dimensions.
Warm Up
Electric Field Hockey
5 minutes
The game you are about to play will allow you to see the behaviour of the electric field, as
well as its relationship to other variables.
Material
●
laptop, tablet, or smartphone
Procedure
1. Open the link below. A screenshot of the game is shown in Fig. 3.6.1.
Electric Field Hockey
University of Colorado Boulder, “Electric Field Hockey,” PhET
Interactive Simulations,
https://phet.colorado.edu/en/simulation/electric-hockey, last
accessed on March 12, 2020.
3.6. Calculating Electric Field
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Unit 3: Electric Potential
Fig. 3.6.1. Electric field hockey
2. Guide the charged puck into the goal. Try to strategize how to position positive and
negative charges on the rink such that when you release the puck, electric forces will
guide it into the goal.
Guide Questions
1. In which direction do field lines point for the positive charges? How about for the
negative charges?
2. What do the field lines signify for the positively-charged puck?
3. What happened when you increased the mass of the puck?
Learn about It!
Plenty of essential conclusions may be drawn from the digital game that you have just
played. Electric field hockey allows you to visualize that the amount of charge plays a
significant role in the electric field, such that even a slight increase or decrease in it will
make the puck change its direction. This lesson provides a mathematical approach to
understanding the concept of the electric field.
3.6. Calculating Electric Field
3
Unit 3: Electric Potential
What are the different ways to calculate electric
field?
Coulomb’s Law and the Electric Field
Suppose a test charge q0 is found at a given location, r, and a system of external charges
acts to exert a force, F on this test charge. The electric field at r is given by:
Equation 3.6.1
where E is the electric field, F is the electrostatic force, and q0 is the test charge.
The electric field at any given region in space is a vector that follows the direction of the
electric force on a test charge at that location, with a magnitude equivalent to the force per
unit charge. It carries the SI unit N/C.
The electric field’s vector value is dependent only on the locations and values of the
external charges. This is supported by Coulomb’s Law, which underscores that the electric
force on a significantly small test charge q0 is directly proportional to the magnitude of the
charge. From Equation 3.6.1, we can derive the electric force experienced by the test
charge at r by:
Remember
A test charge is a charge considered to be infinitesimally small that
it approaches zero. It is assumed that it does not exert any force on
the other charges on the electric field.
3.6. Calculating Electric Field
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Unit 3: Electric Potential
Electric Field of a Single Point Charge
The electric field is considered as a vector. Coulomb’s Law further highlights that the
electric field due to a charge
at the location
is mathematically expressed as:
Equation 3.6.2
where E is the electric field, k is the value of Coulomb’s constant, or 9.0 ✕ 109 Nm2/C2, r is the
distance of the electric field from the charge q; and
is the unit vector that points in the
direction of r.
How can you calculate the electric field due to a
single point charge?
Electric Field at a Point between Two Charges
If the electric field
fields, designated as
at point
,
is due to more than one point charge, the summation of the
,
, etc., due to the charge is determined by getting the total
electric field, as shown in Fig. 3.6.2. This is called the superposition principle, and is given
by:
Applying this principle to the general formula given in Equation 3.6.2, we therefore get:
Equation 3.6.3
3.6. Calculating Electric Field
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Unit 3: Electric Potential
Fig. 3.6.2. Superposition of electric fields
Two-Dimensional Electric Fields
For two-dimensional electric fields, due to two charges q1 and q2 situated at the x- and yaxes, respectively, determine the magnitude of the electric field produced by each of the
charges, then add their components. Since the direction is identified physically, the signs of
the charges are not taken into consideration. Thus,
To determine the direction of the electric field, designate the positive directions as heading
either to the right (positive x-direction) or upward (positive y-direction) and the
negative ones as heading either to the left (negative x-direction) or downward (negative
y-direction). Always remember to consider the attraction and repulsion of like and unlike
charges when considering the direction of the electric field.
Since there are only two forces acting in the x- and y-directions, the magnitude of the
resultant force can be solved with the Pythagorean theorem, thus:
Equation 3.6.4
where ER is the magnitude of the resultant force, EX is the x-component, and EY is the
y-component.
3.6. Calculating Electric Field
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Unit 3: Electric Potential
Moreover, the angle θR can be solved using a trigonometric function, as follows:
Equation 3.6.5
Remember
Remember that by definition,
. In
determining the x- and y-components of the electric field, it is
crucial to first determine the location of the x-component to the
angle θ. If either the x- or y-component is adjacent to this angle,
use the cosine function. On the contrary, if the x- or y-component is
opposite the angle θ, use the sine function.
How are electric fields calculated for charges in
one and in two dimensions?
Let’s Practice!
Example 1
A distance of 11.16 cm separates two point charges, q1, which has a charge of ‒33.13 μC and
q2, which has a charge of 44.15 μC. Identify the field’s magnitude at point A between the two
charges, if A is 3.30 cm away from the negative charge.
Solution
Step 1:
Identify what is required in the problem.
You are asked to calculate the direction and magnitude of the electric field.
3.6. Calculating Electric Field
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Unit 3: Electric Potential
Step 2:
Identify the given in the problem.
The distance between the charges 11.16 cm, the charge of q1, ‒33.13 μC, the
charge of q2, 44.15 μC, and the distance of Point A from q1, 3.30 cm are given.
Determine r2 by subtracting 11.16 cm (the distance between the two charges) and
r1, thus r2 is equal to 7.86 cm.
The electric field diagram illustrating the given values is shown in the figure below.
Step 3:
Express the equation to be used.
Step 4:
Substitute the given values.
Step 5:
Find the answer.
Thus, the electric field is equal to 3.38 ✕ 108 N/C.
3.6. Calculating Electric Field
8
Unit 3: Electric Potential
1 Try It!
Two point charges, q1, which has a charge of ‒76.11 μC and q2 , which has a charge of
13.33 μC, are separated by a distance of 24.67 cm. Identify the field’s magnitude at
point A between the two charges if A is 10.39 cm away from the negative charge.
Example 2
What will be the charge of q1 if an 18-μC point charge is separated from it by a distance of
66.19 cm? Assume that the electric field between the two charges at point A 21-cm away
eastward of q1 is equal to 4.67 ✕ 108 N/C.
Solution
Step 1:
Identify what is required in the problem.
You are asked to calculate the charge of q1.
Step 2:
Identify the given in the problem.
The charge of q2, 18 μC, the distance between q1 and q2, 66.19 cm, the distance
between point A and q1, 21 cm, and the total electric field at point A, 4.67✕108 N/C
are given.
Determine r2 by subtracting 66.19 cm (the distance between the two charges) and
r1, which is 21 cm. Thus, r2 is equal to 45.19 cm.
The electric field diagram illustrating the given values is provided below. Note that
since it was provided that the electric field is found eastward of the first charge,
the value of the charge must be negative.
3.6. Calculating Electric Field
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Unit 3: Electric Potential
Step 3:
Express the equation to be used.
Derive the equation to find q1 as follows:
Step 4:
Substitute the given values.
Step 5:
Find the answer.
Thus, the charge of q1 is equal to 2.28 ✕ 10-3 C.
2 Try It!
What will be the charge of q1 if an 11.90 μC point charge is separated from it by a
distance of 79.10 cm? Assume that the electric field at Point A 34.66-cm away to the
right of q1 is equal to 32.90 ✕ 108 N/C.
3.6. Calculating Electric Field
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Unit 3: Electric Potential
Example 3
Two charges with Point A at their origin form a right triangle, as shown in the figure below.
Their corresponding charges are q1 = 10 μC and q2 = ‒20 μC. The distance between q1 and
point A is 46 mm and the distance between q2 and point A is 14 mm. Determine the
resulting magnitude and direction of the electric field at point A from the two charges.
Solution
Step 1:
Identify what is required in the problem.
You are asked to calculate the electric field at Point A from the two charges.
Step 2:
Identify the given in the problem.
The magnitudes of the two charges, q1 = 10 μC and q2 = ‒20 μC, the
distance between q1 and Point A, 46 mm, and the distance between q2 and
Point A, 14 mm are given.
Step 3:
Express the equations to be used.
3.6. Calculating Electric Field
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Unit 3: Electric Potential
To determine the direction of the electric field, use:
Step 4:
Substitute the given values.
Determine the x- and y-components.
Determine the direction of the electric field.
Step 5:
Find the answers.
Thus, the magnitude of the electric field at point A from the two charges is equal to 9.19
✕ 108 N/C at an angle of 89.3° with respect to the horizontal.
3.6. Calculating Electric Field
12
Unit 3: Electric Potential
3 Try It!
Two charges with Point A at their origin form a right triangle. Their corresponding
charges are q1 = 34.33 μC and q2 = ‒45.78 μC. The distance between q1 and point A is
12 cm and the distance between q2 and point A is 10.11 cm. Determine the resulting
magnitude and direction of the electric field at point A from the two charges.
Key Points
___________________________________________________________________________________________
●
Coulomb’s Law proves that when a test charge q0 is found at a given location, r, and a
system of external charges act to exert a force, F on this test charge, the electric
field at r is equivalent to the amount of force per unit charge.
●
The electric field is expressed through the SI unit newton per coulomb, N/C.
●
A test charge is a charge so small that it does not exert any force on the other
charges on the electric field.
___________________________________________________________________________________________
Key Formulas
___________________________________________________________________________________________
Concept
Formula
Electric Field
Description
Use this equation to
determine the electric field
where
with respect to the electric
●
E is the electric field;
force exerted on a tiny test
●
F is the electrostatic force,
charge.
and
●
3.6. Calculating Electric Field
q0 is the test charge.
13
Unit 3: Electric Potential
Electric Field of a
Use this formula to
Single Point Charge
determine the electric field
where
of a single point charge.
●
E is the electric field;
●
k is the value of Coulomb’s
constant,
or
9
✕
109
Nm2/C2;
●
r is the distance of the
electric
field
from
the
charge q, and
is the unit vector that
●
points in the direction of r.
The Superposition
Use this equation to
Principle
calculate the electric field
between two charges.
where
●
E is the electric field;
●
k is the value of Coulomb’s
constant,
or
9
✕
109
Nm2/C2;
●
q1 and q2 are the charges;
●
r1 is the distance of q2 from
the field, and
●
r2 is the distance of q1 from
the field.
Two-Dimensional
Electric Field
Use this formula when two
where
●
ER is the total electric field;
●
Exis the electric field along
charges are found along
the x- and y-axes,
respectively.
the x-axis, and
3.6. Calculating Electric Field
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Unit 3: Electric Potential
●
Ey is the electric field along
the y-axis.
___________________________________________________________________________________________
Check Your Understanding
A. Identify whether each statement is true or false.
_________________________
1. A test charge is a charge that exerts force on the other
charges on the field.
_________________________
2. The
electric
field
of
a
single
mathematically expressed as
_________________________
point
charge
is
.
3. In the case of two charges, the total electric field at a
given point is equal to the difference of E1 and E2.
_________________________
4. In the equation for calculating two-dimensional field,
,
refers to the distance
between the two charges.
_________________________
5. The cosine function of an arbitrary angle θ can be
determined by the ratio of the hypotenuse of a triangle
to its adjacent side.
_________________________
6. In calculating the electric field at a point between two
charges, it is essential for us to determine the
difference between the electric field of the first charge
and that of the second charge.
3.6. Calculating Electric Field
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Unit 3: Electric Potential
_________________________
7.
The signs of the charges need not to be taken into
consideration in the case of a two-dimensional electric
field.
_________________________
8.
_________________________
9.
Rightward
and
upward
are
considered negative
directions.
When there are only two forces acting in the x- and
y-directions, the magnitude of the resultant force may
be solved using the Pythagorean theorem.
_________________________
10.
The angle
function
may be solved using the trigonometric
.
B. Analyze the problem below and answer the questions that follow.
Two positive charges q1 and q2, both of equal charge, 2.5 μC, are separated at a distance
2 meters.
1. Draw the electric field diagram containing all the given variables and their
respective values.
2. Determine the values of r1 and r2.
3. Determine the value of E1.
4. Determine the value of E2.
5. Calculate the magnitude of the resulting electric field midway between the two
charges.
C. Analyze the problem below and answer the questions that follow.
3.6. Calculating Electric Field
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Unit 3: Electric Potential
Two charges, q1, with a charge of 10 ✕ 10-9 C and q2, with a charge of 5 ✕ 10-9 C are
located along the x- and y-axes, respectively. q1 is 2 ✕ 10-2 m away from the origin, point
P, while q2 is 10 ✕ 10-2 m away from the same point of origin.
1. Draw the electric field diagram containing all the given variables and their
respective values.
2. Determine the value of E1.
3. Determine the value of E2.
4. Calculate the magnitude of the resulting electric field at point P.
5. Find the direction of the electric field.
Challenge Yourself
Take into consideration the formulae you have learned
about calculating electric field, and briefly answer the
following questions.
1. Why must “test charges” used to determine electric fields be assumed to be
extremely small?
2. What determines the strength of an electric field?
3. How are the calculations for electric field related to Gauss’s law?
4. Keeping in mind what you have learned about the potential of conductors, argue why
the electric field inside a conductor is equivalent to zero.
5. When calculating the electric field, can we use both positive and negative charges?
Explain your answer.
Bibliography
3.6. Calculating Electric Field
17
Unit 3: Electric Potential
Faughn, Jerry S. and Raymond A. Serway. Serway’s College Physics (7th ed). Singapore:
Brooks/Cole, 2006.
Giancoli, Douglas C. Physics Principles with Applications (7th ed). USA: Pearson Education,
2014.
Hewitt, Paul G. Conceptual Physics (11th ed). New York: Pearson Education, 2010.
Serway, Raymond A. and John W. Jewett, Jr. Physics for Scientists and Engineers with Modern
Physics (9th ed). USA: Brooks/Cole, 2014.
Young, Hugh D., Roger A. Freedman, and A. Lewis Ford. Sears and Zemansky’s University
Physics with Modern Physics (13th ed). USA: Pearson Education, 2012.
Key to Try It!
1. E = 6.54 ✕ 107 N/C
2. q = 4.39 ✕ 10-2 C
3. E = 4.57 ✕ 107 N/C, at an angle 62.0° with respect to the horizontal
3.6. Calculating Electric Field
18