Electrons are accelerated to reach a kinetic energy of 250 eV (= 4.0 x 10-17 J). If the acceleration is done in a distance of 0.20 m, what is the magnitude of the electric field causing the acceleration?
KE= m•v²/2 => v²= 2•(KE)/m
a=v²/2•s =2(KE)/2•m•s = KE/m•s
m•a=e•E
E= m•a/e = m•KE/e•m•s = KE/e•s =
=250•1.6•10⁻¹⁹/1.6•10⁻¹⁹•0.2 = 1250 V/m
Well, if electrons could talk, they would definitely say it was quite electrifying to reach that kinetic energy! Now, let's compute the magnitude of the electric field causing all that excitement.
The formula to calculate electric field (E) is:
E = √(2qV / m)
Where:
- q is the charge of the electron (around -1.6 x 10^-19 C)
- V is the voltage, which is the kinetic energy in joules (4.0 x 10^-17 J)
- m is the mass of the electron (around 9.1 x 10^-31 kg)
Plugging in those values, we get:
E = √(2 * -1.6 x 10^-19 C * 4.0 x 10^-17 J / 9.1 x 10^-31 kg)
Calculating that gives us approximately:
E ≈ 2.0 x 10^5 N/C
So, the magnitude of the electric field causing the acceleration is approximately 2.0 x 10^5 N/C. That's quite shocking, right?
To find the magnitude of the electric field causing the acceleration, we can use the formula for the work done by an electric field on a charged particle:
W = q * ΔV
where W is the work done, q is the charge of the particle, and ΔV is the change in electric potential.
In this case, the work done is equal to the change in kinetic energy of the electron, which is 4.0 x 10^-17 J.
The charge of an electron is q = 1.6 x 10^-19 C.
So we can rewrite the formula as:
4.0 x 10^-17 J = (1.6 x 10^-19 C) * ΔV
Simplifying the equation, we find:
ΔV = (4.0 x 10^-17 J) / (1.6 x 10^-19 C)
Now, we need to find the electric field (E) using the equation:
E = ΔV / d
where d is the distance over which the acceleration is done. In this case, d = 0.20 m.
Substituting the values into the equation, we get:
E = (4.0 x 10^-17 J) / (1.6 x 10^-19 C) / (0.20 m)
Calculating this expression, we find the magnitude of the electric field causing the acceleration to be:
E ≈ 1.25 x 10^19 N/C
To find the magnitude of the electric field causing the acceleration of the electrons, we can use the formula:
ΔKE = q * ΔV
Where:
ΔKE is the change in kinetic energy of the electrons,
q is the charge of the electron,
and ΔV is the change in voltage.
The change in kinetic energy (ΔKE) is given as 4.0 x 10^(-17) J, and the charge (q) of an electron is 1.6 x 10^(-19) C.
To find the change in voltage (ΔV), we can use the equation of electric potential (V) as:
ΔV = -Ed
Where:
E is the magnitude of the electric field,
and d is the distance over which the acceleration occurs.
Given that ΔV is the change in voltage and d is 0.20 m, we can rearrange the equation to solve for E:
E = -ΔV / d
Now we can substitute the given values into the equation to find the magnitude of the electric field (E):
E = -(4.0 x 10^(-17) J) / (0.20 m)
E = -2.0 x 10^(-16) J/m
Therefore, the magnitude of the electric field causing the acceleration is 2.0 x 10^16 J/m.