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The magnetic field at the origin due to ...

The magnetic field at the origin due to a current element `I vec(dl)` placed at position `r` is
(i) `((mu_(0)i)/(4pi))((dvec(l)xxvec(r))/(r^(3)))`
`-((mu_(0)i)/(4pi))((dvec(l)xxvec(r))/(r^(3)))`
(iii) `((mu_(0)i)/(4pi))((vec(r)xxdvec(l))/(r^(3)))`
`-((mu_(0)i)/(4pi))((vec(r)xxdvec(l))/(r^(3)))`

A

`(mu_(0) I d vec(l) xx vec(r ))/(4 pi r^(2))`

B

`(mu_(0) I vec(r ) xx d vec(l))/(4 pi r^(2))`

C

`(mu_(0) I vec(r ) xx d vec(l))/(4 pi r^(2))`

D

`- (mu_(0) I d vec(l) xx vec(r ))/(4pi r^(2))`

Text Solution

AI Generated Solution

The correct Answer is:
To find the magnetic field at the origin due to a current element \( I \, d\vec{l} \) placed at position \( \vec{r} \), we can use the Biot-Savart law, which states that the magnetic field \( d\vec{B} \) due to a small current element is given by: \[ d\vec{B} = \frac{\mu_0}{4\pi} \frac{I \, d\vec{l} \times \vec{r}}{r^3} \] where: - \( \mu_0 \) is the permeability of free space, - \( I \) is the current, - \( d\vec{l} \) is the vector length of the current element, - \( \vec{r} \) is the position vector from the current element to the point where the magnetic field is being calculated, - \( r \) is the magnitude of the vector \( \vec{r} \). ### Step-by-Step Solution: 1. **Identify the Variables**: - Let \( d\vec{l} \) be the vector representing the current element. - Let \( \vec{r} \) be the position vector from the current element to the origin. 2. **Apply the Biot-Savart Law**: - According to the Biot-Savart law, the magnetic field at the origin due to the current element is given by: \[ d\vec{B} = \frac{\mu_0}{4\pi} \frac{I \, d\vec{l} \times \vec{r}}{r^3} \] 3. **Consider the Direction of the Cross Product**: - The direction of the magnetic field is determined by the right-hand rule applied to the cross product \( d\vec{l} \times \vec{r} \). - If we reverse the order of the vectors in the cross product, we introduce a negative sign: \[ d\vec{B} = -\frac{\mu_0}{4\pi} \frac{\vec{r} \times d\vec{l}}{r^3} \] 4. **Identify the Correct Options**: - From the derived expressions, we can see that the magnetic field can be expressed in two forms: - \( \frac{\mu_0 I}{4\pi} \frac{d\vec{l} \times \vec{r}}{r^3} \) - \( -\frac{\mu_0 I}{4\pi} \frac{\vec{r} \times d\vec{l}}{r^3} \) - Therefore, the correct answers from the options provided are: - \( \frac{\mu_0 I}{4\pi} \frac{d\vec{l} \times \vec{r}}{r^3} \) (Option i) - \( -\frac{\mu_0 I}{4\pi} \frac{\vec{r} \times d\vec{l}}{r^3} \) (Option iv) ### Final Answer: The magnetic field at the origin due to the current element \( I \, d\vec{l} \) placed at position \( \vec{r} \) is given by: - \( \frac{\mu_0 I}{4\pi} \frac{d\vec{l} \times \vec{r}}{r^3} \) (Option i) - \( -\frac{\mu_0 I}{4\pi} \frac{\vec{r} \times d\vec{l}}{r^3} \) (Option iv)
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