Relationship between magnitude of current and magnetic field

[ellieee]

Homework Statement

why is the magnitude of magnetic force acting on both conductors the same even if the currents are not equal? in my opinion, when there is more current, doesn't it mean there is more electricity, so larger magnetic field?

Relevant Equations

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[Doc Al]

Homework Statement:: why is the magnitude of magnetic force acting on both conductors the same even if the currents are not equal? in my opinion, when there is more current, doesn't it mean there is more electricity, so larger magnetic field?

I assume you are talking about the force between two current-carrying conductors.

The conductor with the greater current will produce the greater magnetic field. But the force it exerts on the other conductor is proportional to that second conductor's current. So the force they exert on each other is the same.

 

[Delta2]

Without going into math details, we can invoke Newton's 3rd Law and get that the two forces are equal and opposite, but anyway:
Well let me do the math for you to see that they are equal:
Suppose we have to infinite parallel conductors with currents $I_1,I_2$, and the respective magnetic fields $B_1(r)=\frac{\mu_0}{2\pi}\frac{I_1}{r}$, $B_2(r)=\frac{\mu_0}{2\pi}\frac{I_2}{r-d}$, where d is their in between distance.

Then the force from 1 to 2 $$F_{12}=B_1(d)I_2L$$ while the force from 2 to 1 is $$F_{21}=B_2(0)I_1L$$. It is elementary math to work out to see that $$F_{12}=-F_{21}$$.

 

[ellieee]

But the force it exerts on the other conductor is proportional to that second conductor's current

then example if the "magnitude" of the 1st conductor has a bigger magnetic strength of 10 and the 2nd conductor is 5, so in the end they will exert a magnetic force of magnitude 5 on each other ?

 

[ellieee]

Without going into math details, we can invoke Newton's 3rd Law and get that the two forces are equal and opposite, but anyway:
Well let me do the math for you to see that they are equal:
Suppose we have to infinite parallel conductors with currents $I_1,I_2$, and the respective magnetic fields $B_1(r)=\frac{\mu_0}{2\pi}\frac{I_1}{r}$, $B_2(r)=\frac{\mu_0}{2\pi}\frac{I_2}{r-d}$, where d is their in between distance.

Then the force from 1 to 2 $$F_{12}=B_1(d)I_2L$$ while the force from 2 to 1 is $$F_{21}=B_2(0)I_1L$$. It is elementary math to work out to see that $$F_{12}=-F_{21}$$.

hmmm sorry this Is how it looks like on my phone

 

[Delta2]

Press the refresh page button of your browser.

 

[ellieee]

Without going into math details, we can invoke Newton's 3rd Law and get that the two forces are equal and opposite, but anyway:
Well let me do the math for you to see that they are equal:
Suppose we have to infinite parallel conductors with currents $I_1,I_2$, and the respective magnetic fields $B_1(r)=\frac{\mu_0}{2\pi}\frac{I_1}{r}$, $B_2(r)=\frac{\mu_0}{2\pi}\frac{I_2}{r-d}$, where d is their in between distance.

Then the force from 1 to 2 $$F_{12}=B_1(d)I_2L$$ while the force from 2 to 1 is $$F_{21}=B_2(0)I_1L$$. It is elementary math to work out to see that $$F_{12}=-F_{21}$$.

hmm I don't really get it tho sorry:( can u look at my post 4?

 

[Delta2]

Well it is because if we do the math (which you don't understand but ok I can't do much about it) we end up that both forces are proportional to the product of currents $I_1I_2$, so it doesn't really matter if the currents are not equal. Both currents contribute to the magnitude of both forces.

 

[Doc Al]

then example if the "magnitude" of the 1st conductor has a bigger magnetic strength of 10 and the 2nd conductor is 5, so in the end they will exert a magnetic force of magnitude 5 on each other ?

No. (As @Delta2 explained.)

The field from conductor #1 is proportional to $I_1$; but the force it exerts on conductor #2 is also proportional to the current in #2 ($I_2$). So the force is proportional to $I_1I_2$.

And if you were to figure out the force on #1 from #2, you'd find it's proportional to $I_2I_1$ -- which is the same as $I_1I_2$.

And, as @Delta2 points out, this had better be the case to satisfy Newton's 3rd law.

You might want to read this: Magnetic Force Between Wires