Show that this equation is homogeneous. PLEASE HELP, relatively simple

[MisterOrange]

I have attempted two questions, see below.
1. The relationship between energy, E, and momentum, p, is E = P^2C^2 + M^2C^4 for a relativistic particle, Show that the equation is homogeneous

2. The current, I, in a wire is given by I = nAev where n is the number of electrons per unit volume, A is the cross sectional area of the wire, e is the charge on an electron and v is the drift velocity of the electrons. Show that the equation is homogeneous.

Homework Equations

3.My attempts in photo.

I am assuming 16 is definitely wrong.

and for the questions, so you can see them more articulately.

 

[MisterOrange]

16 is definitely wrong.

I have since worked it.

I = A
n = m^-3
A = m^2
e = C = A*s
V = m*s^-1

which gives you A = m^2 * A * s * m * s^-1 * m^-3

Can anyone help with 17? I don't even know where to start...

 

[gneill]

Question 15 seems to have a typo. The correct expression for the relationship between energy and momentum should have the E squared: E² = ...

So you definitely won't be able to demonstrate that the given equation is homogenous unless this typo is fixed.

However, for the work you've show on question 15 there are some algebra issues. Each term should reduce to units the same as for energy squared. In the fourth line of your solution you "lost" the square on the momentum when you substituted in the units for momentum.

For question 16 you've made the units of n to be C/m³, but it should be just m^-3 as you wrote in your summary.

 

[willem2]

16 is definitely wrong.

I have since worked it.

I = A
n = m^-3
A = m^2
e = C = A*s
V = m*s^-1

which gives you A = m^2 * A * s * m * s^-1 * m^-3

Can anyone help with 17? I don't even know where to start...

16 is ok. you're left with A = A .

17 is just using the same equation as 16. You'll have to convert all the length units to m^3 before you can use the equation, and then convert the speed you get to micrometer/s.
aC is an attocoulomb, which I've never seen used before.

 

[Nickie]

1. To show that an equation is homogeneous, we need to prove that it satisfies the property of homogeneity, which means that if we multiply all variables by a constant, the equation remains unchanged. In other words, the equation is independent of the units used to measure the variables.

In the given equation, we have E = P^2C^2 + M^2C^4. Let's multiply both E and P by a constant, k. This gives us:

kE = (kP)^2C^2 + M^2C^4

Since k is a constant, we can rewrite it as k^2, giving us:

k^2E = (kP)^2C^2 + M^2C^4

Now, we can see that the equation is still the same, even after multiplying the variables by a constant. This proves that the equation is homogeneous.

2. Similarly, for the second equation, we have I = nAev. Let's multiply all variables by a constant, k. This gives us:

kI = knAev

Since k is a constant, we can rewrite it as k^1, giving us:

k^1I = (k^1n)(k^1A)e(k^1v)

Again, we can see that the equation is still the same, even after multiplying the variables by a constant. This proves that the equation is homogeneous.

Overall, both equations satisfy the property of homogeneity, and therefore, can be considered as homogeneous equations.