The question in the image involves finding the work done by a given vector field $\vec{F}$ on a specific curve. I'll solve the problem in full detail step by step in English.

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Problem Statement

We are tasked to find the work done by the vector field $\vec{F}$ along a circular path. The vector field is given as:

$\vec{F} = \left(-\frac{a}{r} + \frac{r}{a}\right)\hat{r} + \frac{r}{a}\sin(\theta)\hat{\phi},$

where:

• $a$ is a constant,
• $r$ is the radial coordinate,
• $\theta$ and $\phi$ are spherical coordinates.

The path is a circular loop in the $xy$-plane (centered at the origin, at $z = 0$) with radius $a$. We will compute the work done along this path.

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Solution

Step 1: General Expression for Work

The work $W$ done by the vector field $\vec{F}$ along a curve $C$ is given by:

$W = \int_C \vec{F} \cdot d\vec{r}.$

Here:

• $\vec{F}$ is the vector field,
• $d\vec{r}$ is the differential element of the curve in the direction of motion,
• $\cdot$ represents the dot product.

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Step 2: Parametrize the Circular Path

The circular loop lies in the $xy$-plane ($z = 0$) at $r = a$. We use the following parametrization for the curve $C$:

$x = a\cos\phi, \quad y = a\sin\phi, \quad z = 0,$

where $\phi \in [0, 2\pi]$.

The differential path element in spherical coordinates is:

$d\vec{r} = a \, d\phi \, \hat{\phi}.$

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Step 3: Evaluate $\vec{F} \cdot d\vec{r}$

The vector field $\vec{F}$ at $r = a$ becomes:

$\vec{F} = \left(-\frac{a}{a} + \frac{a}{a}\right)\hat{r} + \frac{a}{a}\sin(\theta)\hat{\phi}.$

Simplify: $\vec{F} = 0\hat{r} + \sin(\theta)\hat{\phi}.$

At the $xy$-plane, $\theta = \frac{\pi}{2}$, so $\sin(\theta) = 1$. Thus:

$\vec{F} = \hat{\phi}.$

The dot product becomes:

$\vec{F} \cdot d\vec{r} = \hat{\phi} \cdot (a \, d\phi \, \hat{\phi}) = a \, d\phi.$

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Step 4: Compute the Integral

Now, calculate the work:

$W = \int_C \vec{F} \cdot d\vec{r} = \int_0^{2\pi} a \, d\phi.$

Solve the integral:

$W = a \int_0^{2\pi} d\phi = a [\phi]_0^{2\pi} = a (2\pi - 0) = 2\pi a.$

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Final Answer

The work done by the vector field along the circular path is:

$\boxed{W = 2\pi a}.$

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Let me know if you have questions or need clarification!

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Additional Questions and Tips:

1. Why is the radial component of $\vec{F}$ irrelevant for the work calculation?
2. What happens if the radius of the circular loop is not equal to $a$?
3. What role does the symmetry of the vector field play in simplifying the calculation?
4. How would the result change if the path was along a different curve, such as a spiral?
5. What are the general conditions under which the work done by a vector field is zero?

Tip: Always parametrize the curve carefully and confirm it matches the geometry of the problem before integrating!