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Recently I've been studying for my electromagnetism finals and I reached a question about magnetic vector potentials. If I have a wire with constant current distribution, what is the magnetic vector potential inside and outside the wire.

For simplicity sake, I will only include the case of outside the wire. Using Ampere's Law, I have found that the magnetic field at a distance $r$ away from the wire is

$$B=\frac{\mu_{0}I_{total}}{2\pi r}$$

The first method I attempted was to use

$$\oint{\vec{A}}{d\vec{l}}=\int{\vec{B}}{d\vec{S}}$$ In this case I am taking a closed loop with radius equal to $r$. $\oint{d\vec{l}}$ should just give the circumference and since $\vec{S}$ is $\pi r^{2}$, $d\vec{S}$ should be $2\pi rd\vec{r}$.

Or at least I think it is so, because if I plug in these values, I get that the magnetic vector potential outside is a constant.

$$\vec{A}(2\pi r)=\int_{0}^{r}{\frac{\mu_{0}I_{tot}}{2\pi r'}2\pi r'dr'}$$ $$\vec{A}=\frac{\mu_{0}I_{total}}{2\pi}$$

The second method I attempted was to say that

$$\vec{B}=\nabla\times\vec{A}$$ And since $\vec{B}$ is in the $\hat{\phi}$ direction, and assuming $\vec{A}$ only varies in the $\hat{r}$ direction, doing the cross product just results in

$$\frac{\mu_{0}I_{total}}{2\pi r}=-\frac{\partial A_{z}}{\partial r}$$

Which after integrating gives me a logarithm.

Doing research online, I found that the magnetic vector potential outside of a wire is supposed to give a logarithm, which means my first method is flawed. The problem is I don't know where. I feel like I'm misunderstanding something about the vectors involved. Any help would be appreciated. Thank you!

Recently I've been studying for my electromagnetism finals and I reached a question about magnetic vector potentials. If I have a wire with constant current distribution, what is the magnetic vector potential inside and outside the wire.

For simplicity sake, I will only include the case of outside the wire. Using Ampere's Law, I have found that the magnetic field at a distance $r$ away from the wire is

$$B=\frac{\mu_{0}I_{total}}{2\pi r}$$

The first method I attempted was to use

$$\oint{\vec{A}}{d\vec{l}}=\int{\vec{B}}{d\vec{S}}$$ In this case I am taking a closed loop with radius equal to $r$. $\oint{d\vec{l}}$ should just give the circumference and since $\vec{S}$ is $\pi r^{2}$, $d\vec{S}$ should be $2\pi rd\vec{r}$.

Or at least I think it is so, because if I plug in these values, I get that the magnetic vector potential outside is a constant.

The second method I attempted was to say that

$$\vec{B}=\nabla\times\vec{A}$$ And since $\vec{B}$ is in the $\hat{\phi}$ direction, and assuming $\vec{A}$ only varies in the $\hat{r}$ direction, doing the cross product just results in

$$\frac{\mu_{0}I_{total}}{2\pi r}=-\frac{\partial A_{z}}{\partial r}$$

Which after integrating gives me a logarithm.

Doing research online, I found that the magnetic vector potential outside of a wire is supposed to give a logarithm, which means my first method is flawed. The problem is I don't know where. I feel like I'm misunderstanding something about the vectors involved. Any help would be appreciated. Thank you!

Recently I've been studying for my electromagnetism finals and I reached a question about magnetic vector potentials. If I have a wire with constant current distribution, what is the magnetic vector potential inside and outside the wire.

For simplicity sake, I will only include the case of outside the wire. Using Ampere's Law, I have found that the magnetic field at a distance $r$ away from the wire is

$$B=\frac{\mu_{0}I_{total}}{2\pi r}$$

The first method I attempted was to use

$$\oint{\vec{A}}{d\vec{l}}=\int{\vec{B}}{d\vec{S}}$$ In this case I am taking a closed loop with radius equal to $r$. $\oint{d\vec{l}}$ should just give the circumference and since $\vec{S}$ is $\pi r^{2}$, $d\vec{S}$ should be $2\pi rd\vec{r}$.

Or at least I think it is so, because if I plug in these values, I get that the magnetic vector potential outside is a constant.

$$\vec{A}(2\pi r)=\int_{0}^{r}{\frac{\mu_{0}I_{tot}}{2\pi r'}2\pi r'dr'}$$ $$\vec{A}=\frac{\mu_{0}I_{total}}{2\pi}$$

The second method I attempted was to say that

$$\vec{B}=\nabla\times\vec{A}$$ And since $\vec{B}$ is in the $\hat{\phi}$ direction, and assuming $\vec{A}$ only varies in the $\hat{r}$ direction, doing the cross product just results in

$$\frac{\mu_{0}I_{total}}{2\pi r}=-\frac{\partial A_{z}}{\partial r}$$

Which after integrating gives me a logarithm.

Doing research online, I found that the magnetic vector potential outside of a wire is supposed to give a logarithm, which means my first method is flawed. The problem is I don't know where. I feel like I'm misunderstanding something about the vectors involved. Any help would be appreciated. Thank you!

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Inconsistencies in finding magnetic vector potentials

Recently I've been studying for my electromagnetism finals and I reached a question about magnetic vector potentials. If I have a wire with constant current distribution, what is the magnetic vector potential inside and outside the wire.

For simplicity sake, I will only include the case of outside the wire. Using Ampere's Law, I have found that the magnetic field at a distance $r$ away from the wire is

$$B=\frac{\mu_{0}I_{total}}{2\pi r}$$

The first method I attempted was to use

$$\oint{\vec{A}}{d\vec{l}}=\int{\vec{B}}{d\vec{S}}$$ In this case I am taking a closed loop with radius equal to $r$. $\oint{d\vec{l}}$ should just give the circumference and since $\vec{S}$ is $\pi r^{2}$, $d\vec{S}$ should be $2\pi rd\vec{r}$.

Or at least I think it is so, because if I plug in these values, I get that the magnetic vector potential outside is a constant.

The second method I attempted was to say that

$$\vec{B}=\nabla\times\vec{A}$$ And since $\vec{B}$ is in the $\hat{\phi}$ direction, and assuming $\vec{A}$ only varies in the $\hat{r}$ direction, doing the cross product just results in

$$\frac{\mu_{0}I_{total}}{2\pi r}=-\frac{\partial A_{z}}{\partial r}$$

Which after integrating gives me a logarithm.

Doing research online, I found that the magnetic vector potential outside of a wire is supposed to give a logarithm, which means my first method is flawed. The problem is I don't know where. I feel like I'm misunderstanding something about the vectors involved. Any help would be appreciated. Thank you!