Can two variables in two different equations cancel each other?

[Y0SHI0N]

Homework Statement

What is the relationship between the force acting on a current-carrying wire within a magnetic field and the resistivity of the wire material at room temperature?

Relevant Equations

F = BIL sin⁡θ (#1)
I = V/R (#2)
R = ρ l/A (#3)
F = BLVA sin⁡θ/ρl (#4)

Using the related equations, I have come up with an equation that I think would answer this question (#4). However, I was just wondering does the 'L' from equation #1 and 'l' in equation #3 cancels each other out when the substitution is made to give out equation #4 (since they are both the length of the wire)? I asked my physics teacher in class and he didn't know the answer. Thank you so much for your time, every response is much appreciated.

Edit: the teacher just came back to me with a response. He said that the 'l' and 'L' are two different lengths despite the same wire are being used in this instance. The 'L' here represents the length of which the force is acting on the wire based on how much of the wire is going through the magnetic field whereas the 'l' represents the full length of the wire. Can someone please confirm this explanation? Thank you.

 

[haruspex]

Can someone please confirm this explanation? Thank you.

Looks right.

 

[Y0SHI0N]

Looks right.

Say if I were to set up the scenario in question with crocodile and a simple power source, would this relationship change? do I need to consider the resistance of the crocodile cables as well? Also thank you for your response.

 

[haruspex]

Say if I were to set up the scenario in question with crocodile and a simple power source, would this relationship change? do I need to consider the resistance of the crocodile cables as well? Also thank you for your response.

Every equation has a context. It is only valid under a stated set of conditions.
R = ρ l/A is valid for a length of conductor of uniform resistivity and cross sectional area.
I = V/R is valid between two points with potential difference V etc.
For the purposes of the question in post #1, you could make the context just a short section of wire, entirely in the field, and with a voltage V applied across its length.

 

[Y0SHI0N]

Every equation has a context. It is only valid under a stated set of conditions.
R = ρ l/A is valid for a length of conductor of uniform resistivity and cross sectional area.
I = V/R is valid between two points with potential difference V etc.
For the purposes of the question in post #1, you could make the context just a short section of wire, entirely in the field, and with a voltage V applied across its length.

Thank you so much:)

 

[etotheipi]

Generally you cannot assume that the same letter appearing in two different equations you find on the internet represents the same quantity. And you definitely can't assume that a lower-case and upper-case symbol, e.g. $a$ and $A$, appearing in a given term represent the same quantity!