Class 11 > Unit # 05: Work, Energy & Power > Conservative Field

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Work and Energy - Conservative Fields

Q. Define Conservative field and prove that gravitational field is a conservative field.

Conservative Field

"A force field is said to be conservative if the work done in moving a body along any closed path is equal to zero."

Examples:

1. Gravitational Field
2. Electrostatic Field
3. Magnetic Field

Figure: Closed triangular path ABCA in a gravitational field used for proof.

Proof: Gravitational Field is a Conservative Field

Suppose a body is moved along a closed triangular path $\text{ABCA}$ in a gravitational field. To prove that the field is conservative, we calculate the work done along each section of the path: $\text{A} \to \text{B}$, $\text{B} \to \text{C}$, and $\text{C} \to \text{A}$.

1. Work Done from A to B ($W_{\text{A}\to\text{B}}$)

The work done moving from $\text{A}$ to $\text{B}$ is given by:

$W_{\text{A}\to\text{B}} = \vec{F} \cdot \vec{S}_1 = F S_1 \cos\alpha$

In right-angled triangle $\text{BAD}$:

$\cos\alpha = \frac{h}{S_1} \implies S_1 \cos\alpha = h$

Substituting this into the work equation:

$W_{\text{A}\to\text{B}} = Fh = mgh \quad \text{--- (i)}$

2. Work Done from B to C ($W_{\text{B}\to\text{C}}$)

Along path $\text{B}\to\text{C}$, the displacement is horizontal while the gravitational force acts vertically downwards (angle $\theta = 90^\circ$).

$W_{\text{B}\to\text{C}} = \vec{F} \cdot \vec{S}_2 = F S_2 \cos 90^\circ = S_2(0) = 0 \quad \text{--- (ii)}$

3. Work Done from C to A ($W_{\text{C}\to\text{A}}$)

$W_{\text{C}\to\text{A}} = \vec{F} \cdot \vec{S}_3 = F S_3 \cos(180^\circ - \beta) = -F S_3 \cos\beta$

In right-angled triangle $\text{CAD}$:

$\cos\beta = \frac{h}{S_3} \implies S_3 \cos\beta = h$

Substituting this into the work equation:

$W_{\text{C}\to\text{A}} = -Fh = -mgh \quad \text{--- (iii)}$

4. Total Work Done Along the Closed Path ABCA

The total work done in completing the closed loop is the algebraic sum of the work done along each path segment:

$W_{\text{total}} = W_{\text{A}\to\text{B}} + W_{\text{B}\to\text{C}} + W_{\text{C}\to\text{A}}$
$W_{\text{total}} = mgh + 0 + (-mgh) = 0$

Hence, it is proved that the gravitational field is a conservative field.

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