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QPages 275-276: Q8: HC Verma: Concepts of Physics - Volume 2 Permanent Magnets Pages 275-276: Q8 Solution: Magnetic scalar potential is defined as . Apply this equation…

Magnetic scalar potential is defined as Ur⃗2)−Ur⃗1)=−∫r⃗1r⃗2B⃗⋅dl⃗U\vec{r}_2) - U\vec{r}_1) = -\int_{\vec{r}_1}^{\vec{r}_2} \vec{B} \cdot d\vec{l}. Apply this equation to a closed curve enclosing a long straight wire. The RHS of the above equation is then −μ0i-\mu_0 i by Ampere's law. We see that Ur⃗2)≠Ur⃗1)U\vec{r}_2) \neq U\vec{r}_1) even when r⃗2=r⃗1\vec{r}_2 = \vec{r}_1. Can we have a magnetic scalar potential in this case?

Step-by-Step Verified Solution

**Answer:** No, we cannot have a magnetic scalar potential in this case. **Step-by-Step Explanation:** For a closed curve enclosing a current, Ampere's law gives ∮B⃗⋅dl⃗=μ0i≠0\oint \vec{B} \cdot d\vec{l} = \mu_0 i \neq 0. For a scalar potential UU to exist, the line integral around any closed path must be zero. Since it is not zero here, the magnetic scalar potential is not single-valued and cannot be defined globally [citation:3].

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