(1) Dervive an expression for the maximum speed of v of electron in terms of V,me, and e.
(2) The electron is replaced with a proton that starts from rest at the positive plate and travels toward the negative plate. Compare the maximum speed of the protons to the maximum speed of the electron in question 82 and give evidence to support your answer.
(3) according to Ohm's law an ohm is equivalent to a volt per ampere. The ohm can be expressed in terms of the fundamental units kilogram, meter,second, and ampere. Show 1 V/A = 1 kg.m^3/A^2.s^3
(4) Derive an expression for the radius of a conductor in terms of the length, resistance, and resistivity.
Please someone help me with this thank you:)
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(1) To derive the expression for the maximum speed of an electron (v) in terms of V, me, and e, we can start by using the conservation of energy principle.
The electrical potential energy gained by an electron moving in an electric field is given by:
ΔPE = eV
Where e is the charge of the electron and V is the voltage. This electrical potential energy is converted into kinetic energy.
The kinetic energy of an object can be expressed as:
KE = (1/2)mv^2
Where m is the mass of the electron and v is its velocity.
Since the total energy remains constant, we can equate the electrical potential energy with the kinetic energy:
eV = (1/2)mev^2
Simplifying the equation, we can solve for v:
v = sqrt((2eV) / me)
Therefore, the expression for the maximum speed of the electron (v) in terms of V, me, and e is:
v = sqrt((2eV) / me)
(2) When the electron is replaced with a proton, we need to consider the differences in charge and mass.
The charge of a proton (e) is equal in magnitude but opposite in sign to the charge of an electron. The mass of the proton (mp) is significantly greater than the mass of an electron (me).
So, when the proton starts from rest at the positive plate and travels towards the negative plate, the same principles of energy conservation apply.
Using the same equation as before, the maximum speed of the proton can be derived as:
v_p = sqrt((2eV) / mp)
Since mp > me, the mass of the proton is greater than that of the electron, the denominator of the equation for the proton's maximum speed v_p is larger. This means that the proton will have a slower maximum speed compared to the electron.
The evidence supporting this can be found by comparing the derived expressions for the maximum speed of the electron (v) and the proton (v_p). The denominator for the electron (me) is smaller than the denominator for the proton (mp), resulting in a higher maximum speed for the electron.
(3) To show that 1 V/A is equivalent to 1 kg.m^3/A^2.s^3, we can use Ohm's law and express resistance (ohm) in terms of the fundamental units kilogram, meter, second, and ampere.
Ohm's law states:
V = IR
Where V is voltage, I is current, and R is resistance.
From the given information that an ohm is equivalent to a volt per ampere, we have:
1 ohm = 1 V/A
Now, we can substitute R with 1 ohm in Ohm's law equation:
V = I * 1 ohm
Since we want to express 1 V/A in terms of kilogram, meter, second, and ampere, we need to express current (I) in terms of these units.
Current (I) is defined as the flow of charge (Q) over time (t):
I = Q / t
Charge (Q) is expressed as the product of current (I) and time (t):
Q = I * t
Substituting this expression for charge (Q) in the equation for voltage (V):
V = (I * t) * 1 ohm
Simplifying, we can rearrange the equation:
V = I * (1 ohm / 1) * t
Therefore, we can conclude that 1 V/A is equivalent to 1 kg.m^3/A^2.s^3.
(4) To derive an expression for the radius (r) of a conductor in terms of length (L), resistance (R), and resistivity (ρ), we can use the formula for resistance:
R = (ρ * L) / A
Where R is resistance, ρ is resistivity, L is length, and A is the cross-sectional area of the conductor.
To isolate the cross-sectional area A, we can rearrange the equation:
A = (ρ * L) / R
The formula for the cross-sectional area of a conductor can be expressed as the product of π (pi) and the square of the radius (r):
A = π * r^2
Now we can equate the two expressions for A:
(ρ * L) / R = π * r^2
To solve for the radius (r), we rearrange the equation:
r^2 = [(ρ * L) / (R * π)]
Taking the square root of both sides:
r = sqrt[(ρ * L) / (R * π)]
Therefore, the expression for the radius (r) of a conductor in terms of length (L), resistance (R), and resistivity (ρ) is:
r = sqrt[(ρ * L) / (R * π)]