An electron moving parallel to the x axis has an initial speed of 4.82 *10^6 m/s at the origin. Its speed is reduced to 1.46 * 10^5 m/s at the point x = 2.00 cm.
Calculate the electric potential difference between the origin and that point.
To calculate the electric potential difference between the origin and the point x = 2.00 cm, we can use the equation:
ΔV = -e * ΔVb
where ΔV is the electric potential difference, e is the charge of an electron (1.6 * 10^-19 C), and ΔVb is the change in kinetic energy of the electron.
The change in kinetic energy can be calculated using the formula:
ΔK.E = (1/2) * m * (vf^2 - vi^2)
where ΔK.E is the change in kinetic energy, m is the mass of the electron (9.11 * 10^-31 kg), vf is the final velocity of the electron, and vi is the initial velocity of the electron.
Given:
vi = 4.82 * 10^6 m/s
vf = 1.46 * 10^5 m/s
m = 9.11 * 10^-31 kg
Let's calculate ΔK.E first:
ΔK.E = (1/2) * 9.11 * 10^-31 * (1.46 * 10^5)^2 - (4.82 * 10^6)^2
Simplifying the expression gives us:
ΔK.E ≈ -2.48 * 10^-17 J
Now, let's calculate ΔV:
ΔV = -1.6 * 10^-19 C * (-2.48 * 10^-17 J)
Simplifying the expression gives us:
ΔV ≈ 3.97 V
Therefore, the electric potential difference between the origin and the point x = 2.00 cm is approximately 3.97 volts.
To calculate the electric potential difference between the origin and a point, we can use the principle of conservation of energy. We know that the only force acting on the electron is the electric force, which does work on the electron to change its kinetic energy.
The electric force acting on a charged particle is given by the equation F = qE, where F is the force, q is the charge of the particle, and E is the electric field strength.
The work done by the electric force on the particle is given by the equation W = qΔV, where W is the work done, q is the charge of the particle, and ΔV is the electric potential difference.
We can equate the work done by the electric force to the change in kinetic energy of the electron:
W = ΔKE
Since ΔKE = KE_final - KE_initial, we have:
qΔV = (1/2)m(v_final^2 - v_initial^2)
where q is the charge of the electron, ΔV is the electric potential difference, m is the mass of the electron, v_final is the final velocity of the electron, and v_initial is the initial velocity of the electron.
First, we need to calculate the charge of the electron. The charge of an electron is -1.6 x 10^-19 Coulombs.
Next, we need to calculate the mass of the electron. The mass of an electron is 9.1 x 10^-31 kilograms.
Now, we can substitute the given values into the equation:
-1.6 x 10^-19 C * ΔV = (1/2) * 9.1 x 10^-31 kg * ((1.46 x 10^5 m/s)^2 - (4.82 x 10^6 m/s)^2)
Simplifying the equation:
-1.6 x 10^-19 C * ΔV = (1/2) * 9.1 x 10^-31 kg * (2.1316 x 10^10 m^2/s^2)
-1.6 x 10^-19 C * ΔV = 1.7147 x 10^-20 J
Finally, solving for ΔV:
ΔV = (1.7147 x 10^-20 J) / (-1.6 x 10^-19 C)
ΔV ≈ -0.107 J/C
Therefore, the electric potential difference between the origin and the point x = 2.00 cm is approximately -0.107 J/C.