1. Coulomb's Law
Coulomb's
Law is a fundamental principle in electrostatics that describes the interaction
between electric charges. The law states that the force between two charged
particles is proportional to the product of their charges and inversely
proportional to the square of the distance between them.
The
mathematical expression of Coulomb's Law is:
F
= k * (q1 * q2) / r^2
where
F is the electrostatic force between two point charges, q1 and q2 are the
magnitudes of the charges, r is the distance between the charges, and k is
Coulomb's constant, a proportionality constant that depends on the medium
between the charges.
Coulomb's
Law can be used to determine the force between two charged particles, the
electric field produced by a charged particle, and the potential energy
associated with the interaction of two charged particles.
Coulomb's
Law is important in many fields, including physics, chemistry, and electrical
engineering. It provides a foundation for understanding the behavior of charged
particles and their interactions with each other and with their environment,
and it has many practical applications in fields such as power systems,
electronics, and telecommunications.
The derivation of Coulomb's Law:
It involves analyzing the forces that exist between
two point charges. Here is a brief outline of the derivation:
Assume
that there are two point charges, q1 and q2, located at positions r1 and r2,
respectively.
The
force on q1 due to q2 can be represented by the vector equation:
F
= k * q1 * q2 * (r2 - r1) / |r2 - r1|^3
where
k is Coulomb's constant, and |r2 - r1| is the distance between the two charges.
By
Newton's third law, the force on q2 due to q1 is equal in magnitude and
opposite in direction to the force on q1 due to q2.
For
simplicity, we can assume that q2 is fixed at some point in space, so that r2
is constant. Then, we can consider the force on q1 due to q2 as a function of
the position r1.
Taking
the gradient of the force with respect to r1, we obtain the electric field
produced by q2 at the position of q1:
E
= k * q2 * (r1 - r2) / |r1 - r2|^3
The
force on q1 due to q2 can be obtained by multiplying the electric field by the
charge of q1:
F
= q1 * E
Substituting
the expression for E obtained in step 5, we arrive at the final form of
Coulomb's Law:
F
= k * q1 * q2 / |r1 - r2|^2
This
derivation shows that Coulomb's Law can be derived from the concept of electric
fields, which are created by charged particles and can exert forces on other
charged particles. The derivation also shows the importance of Coulomb's
constant, which provides a measure of the strength of the electric force
between two charged particles.
Here are few questions with solutions to Coulomb's Law:
Two point charges of +2 C and -3 C are separated by a distance of 10 m. What is the force between them?
Solution: Using
Coulomb's Law, we have F = k * q1 * q2 / r^2, where k is Coulomb's constant.
Plugging in the values, we get F = (9 x 10^9 Nm^2/C^2) * (2 C) * (-3 C) / (10
m)^2 = -5.4 x 10^8 N.
A point charge of +4 μC is located 2 cm away from a point charge of -6 μC. What is the force between them?
Solution: Converting
the distances to SI units, we have r = 0.02 m. Using Coulomb's Law, we have F =
k * q1 * q2 / r^2. Plugging in the values, we get F = (9 x 10^9 Nm^2/C^2) * (4
x 10^-6 C) * (-6 x 10^-6 C) / (0.02 m)^2 = -6.75 x 10^-3 N.
Two point charges of -5 nC and +2 nC are separated by a distance of 1 cm. What is the force between them?
Solution:
Converting the distances to SI units, we have r = 0.01 m. Using Coulomb's Law,
we have F = k * q1 * q2 / r^2. Plugging in the values, we get F = (9 x 10^9
Nm^2/C^2) * (-5 x 10^-9 C) * (2 x 10^-9 C) / (0.01 m)^2 = -9 x 10^-3 N.
A charge of +5 μC is placed at a distance of 4 cm from a charge of -2 μC. What is the electric field at a point midway between them?
Solution: Converting the distances to SI units, we have r = 0.04 m. Using
Coulomb's Law, we have F = k * q1 * q2 / r^2. Plugging in the values, we get F
= (9 x 10^9 Nm^2/C^2) * (5 x 10^-6 C) * (-2 x 10^-6 C) / (0.04 m)^2 = -5.625 N.
The electric field at the midpoint is then E = F / q = -5.625 N / (5 x 10^-6 C)
= -1.125 x 10^6 N/C
The
electric field is a physical quantity used to describe the influence that a
charged object exerts on its surroundings. It is a vector field that describes
the direction and magnitude of the force that a charged object would exert on a
test charge placed at any given point in space. The electric field is created
by charged particles, and it is responsible for many of the phenomena
associated with electricity, such as electric currents, electric potential, and
electric charges. The units of electric field are newtons per coulomb (N/C) or
volts per meter (V/m).
formula
for electric field
The
formula for electric field at a point in space is given by:
E
= F / q
where
E is the electric field, F is the electric force experienced by a test charge q
placed at that point, and q is the magnitude of the test charge.
Mathematically, the electric field is defined as the force per unit charge.
The
formula for the electric field due to a point charge Q at a distance r from the
charge is given by:
E
= kQ / r^2
where
k is Coulomb's constant, equal to approximately 9 × 10^9 N·m^2/C^2. This
formula describes how the electric field strength decreases with distance from
the point charge.
List
of formulae for electric field
Here
are some important formulas related to the electric field:
Electric
field due to a point charge: E = kQ / r^2 where E is the electric field at a
distance r from the point charge Q, and k is Coulomb's constant (approx. 9 ×
10^9 N·m^2/C^2).
Electric
field due to a uniformly charged sphere: E = kQ / R^2 (for r ≤ R) where E is
the electric field at a distance r from the center of the sphere, Q is the
total charge on the sphere, and R is the radius of the sphere.
Electric
field due to an infinite line of charge: E = λ / (2πε0r) where λ is the linear
charge density (charge per unit length) of the line, ε0 is the permittivity of
free space, and r is the distance from the line.
Electric
field due to an infinite sheet of charge: E = σ / (2ε0) where σ is the surface
charge density (charge per unit area) of the sheet, and ε0 is the permittivity
of free space.
Electric
field due to a charged disk: E = (1 / 2ε0) × (σR / √(R^2 + z^2)) where σ is the
surface charge density of the disk, R is the radius of the disk, and z is the
distance from the center of the disk to the point where the electric field is
being measured.
The electric field inside a parallel-plate capacitor: E = V / d where V is the potential
difference between the plates, d is the distance between the plates, and E is
the electric field strength between the plates.
These are just a few of the many formulas related to electric fields. Other formulas exist for more complex charge distributions or arrangements of charged objects.
Here
are few problems related to the electric field:
What
is the electric field at a distance of 2 meters from a point charge of 4
micro-coulombs?
Solution:
Using the formula for electric field due to a point charge, we have E = kQ /
r^2, where k is Coulomb's constant, Q is the charge on the point charge, and r
is the distance from the point charge. Plugging in the values, we get E = (9 ×
10^9 N·m^2/C^2) × (4 × 10^-6 C) / (2 m)^2 = 9 × 10^3 N/C.
Answer:
The electric field at a distance of 2 meters from the point charge is 9,000
N/C.
A
point charge of 8 micro-coulombs is located at the origin. What is the electric
field at a distance of 3 meters on the x-axis?
Solution:
We can use the formula for electric field due to a point charge. The distance
from the point charge to the point on the x-axis is 3 meters, so we have r = 3
meters. Plugging in the values, we get E = (9 × 10^9 N·m^2/C^2) × (8 × 10^-6 C)
/ (3 m)^2 = 2.96 × 10^6 N/C.
Answer:
The electric field at a distance of 3 meters on the x-axis is 2.96 × 10^6 N/C.
What
is the electric field at a point that is 4 meters away from a line charge of 6
micro-coulombs/meter?
Solution:
We can use the formula for electric field due to an infinite line of charge,
which is E = λ / (2πε0r), where λ is the linear charge density of the line, ε0
is the permittivity of free space, and r is the distance from the line charge.
Plugging in the values, we get E = (6 × 10^-6 C/m) / (2π × 8.85 × 10^-12 F/m) ×
(4 m) = 85.8 N/C.
Answer:
The electric field at a point that is 4 meters away from the line charge is
85.8 N/C.
What
is the electric field between two parallel plates with a potential difference
of 100 volts and a plate separation of 2 cm?
Solution:
We can use the formula for the electric field inside a parallel-plate
capacitor, which is E = V / d, where V is the potential difference between the
plates, and d is the distance between the plates. Plugging in the values, we
get E = (100 V) / (0.02 m) = 5000 N/C.
Answer:
The electric field between the two parallel plates is 5000 N/C.
3. Gauss's Law
Gauss's
Law is a fundamental principle of electromagnetism that relates the
distribution of electric charge to the electric field it creates. The law was
discovered by Carl Friedrich Gauss, and it states that the electric flux
through a closed surface is proportional to the total electric charge enclosed
within that surface.
Electric
field due to infinite uniform line charge in gauss law
