two charged particles of masses 4m & m and having charges +q & +3q are placed in uniform electric field they are allowed to move for 2 sec find the ratio of their kinetic energy solution
To find the ratio of the kinetic energies of the two charged particles, we can start by considering the equations of motion.
The equation of motion for an object in a uniform electric field is given by:
F = qE,
where F is the force experienced by the charged particle, q is its charge, and E is the electric field strength.
The force experienced by a charged particle moving in an electric field can be related to its acceleration by Newton's second law:
F = ma,
where m is the mass of the particle and a is its acceleration.
Combining these equations, we have:
qE = ma.
We can rearrange this equation to express acceleration in terms of the force and mass:
a = qE/m.
Now, we know that the particles are allowed to move for 2 seconds. The equation of motion for an object moving with constant acceleration is given by:
s = ut + (1/2)at^2,
where s is the distance traveled, u is the initial velocity, t is the time, a is the acceleration, and we assume the initial velocity u is zero.
Since the particles start from rest (u = 0), the equation simplifies to:
s = (1/2)at^2.
Substituting the expression for acceleration, we have:
s = (1/2)(qE/m)t^2.
The distance traveled by each particle is the same, so we have:
(1/2)(qE/m)(t^2 for particle 1) = (1/2)(qE/m)(t^2 for particle 2).
Cancelling out the common terms, we are left with:
t^2 for particle 1 = t^2 for particle 2.
Taking the square root of both sides, we have:
t for particle 1 = t for particle 2.
Since the time is the same for both particles, we can conclude that the ratio of their kinetic energies is equal to the ratio of their masses:
Kinetic energy for particle 1 / Kinetic energy for particle 2 = mass of particle 1 / mass of particle 2.
Plugging in the given masses of 4m and m, we have:
Kinetic energy for particle 1 / Kinetic energy for particle 2 = 4m / m = 4.
Therefore, the ratio of their kinetic energies is 4.
To find the ratio of the kinetic energy of two charged particles in a uniform electric field, we need to consider the effect of the electric field on the particles' motion.
The electric force experienced by a charged particle in an electric field is given by the equation:
F = qE
Where F is the force experienced, q is the charge of the particle, and E is the electric field strength.
Assuming the particles have positive charges, the forces experienced by the two particles will be in the same direction as the electric field.
Since the particles start from rest, we can use the equation:
F = ma
To relate the force experienced by the particles to their acceleration. Here, m is the mass of the particle and a is the acceleration.
For the first particle with a mass of 4m and a charge of +q, the equation becomes:
qE = (4m)a1
For the second particle with a mass of m and a charge of +3q, the equation becomes:
3qE = ma2
Since both particles are allowed to move for 2 seconds, we can use the kinematic equation:
v = u + at
Where v is the final velocity, u is the initial velocity (which is 0 in this case), a is the acceleration, and t is the time.
Rearranging this equation, we can find the velocity v:
v = at
The kinetic energy (KE) of a particle is given by the equation:
KE = (1/2)mv²
Now, let's find the ratio of the kinetic energies of the two particles.
For the first particle:
Initial velocity u1 = 0
Final velocity v1 = a1 * 2 (since the particle is allowed to move for 2 seconds)
Kinetic energy KE1 = (1/2) * (4m) * (a1 * 2)²
For the second particle:
Initial velocity u2 = 0
Final velocity v2 = a2 * 2 (since the particle is allowed to move for 2 seconds)
Kinetic energy KE2 = (1/2) * m * (a2 * 2)²
To find the ratio, divide the kinetic energy of the second particle (KE2) by the kinetic energy of the first particle (KE1):
Ratio = KE2 / KE1 = [(1/2) * m * (a2 * 2)²] / [(1/2) * (4m) * (a1 * 2)²]
Simplifying the equation, the masses cancel out, and we are left with:
Ratio = [(a2 * 2)²] / [(a1 * 2)²]
= 4(a2²) / 4(a1²)
= (a2²) / (a1²)
Since we don't have specific values for a1 and a2, we are unable to determine a numerical ratio without additional information or further calculations.
In summary, the ratio of the kinetic energies of two charged particles with masses 4m and m, and charges +q and +3q, respectively, in a uniform electric field cannot be determined without knowing the specific values of their accelerations.
The electrostatic force exerted on the charged particle can be given as:
F
=
q
E
.
Therefore, acceleration of the particle is given by:
a
=
F
m
=
q
E
m
.
The final velocity of the charged particle in time
t
is given by:
v
=
a
t
=
q
E
m
t
.
Then kinetic energy of the particle in time
t
is given by:
K
=
1
2
m
v
2
=
1
2
m
×
(
q
E
m
t
)
2
=
q
2
E
2
t
2
2
m
.
Therefore, if the electric field and time are kept constant, then:
K
∝
q
2
m
.
If we consider two charged particles of masses
m
1
and
m
2
and respective charges
q
1
and
q
2
, being accelerated in the same electric field and for the same interval of time, the ratio of their kinetic energies will be given by:
K
1
K
2
=
(
q
1
q
2
)
2
m
1
m
2
K
1
K
2
=
(
q
1
q
2
)
2
m
1
m
2
K
1
K
2
=
(
q
3
q
)
2
4
m
m
K
1
K
2
=
1
36