Chapter 3: Q3.6P (page 124)

A more elegant proof of the second uniqueness theorem uses Green's
identity (Prob. 1.61c), with $T = U = V_{3}$ . Supply the details.

Short Answer

Expert verified

Answer

It is proved as second uniqueness of theorem by using greens identity.

Step by step solution

Define function

Write the Greens identity.

$\int_{~ N} [(T {~ N}^{2} U) + ~ NU \times ~ NT] dτ = \oint (T ~ NU) \times da$ …… (1)

Given that $T = U = V_{3}$ ,

Therefore, the greens identity is changes as,

$\int_{~ N} [(V_{3} {~ N}^{2} V_{3}) + ~ {NV}_{3} \times ~ {NV}_{3}] dτ = \oint V_{3} (~ {NV}_{3}) \times da$ …… (2)

Determine proof of Greens identity

Since

$\nabla^{2} V_{3} = \nabla^{2} V_{1} - \nabla^{2} V_{2} \nabla^{2} V_{1} = \frac{- ρ}{ε_{0}} \nabla^{2} V_{2} = \frac{- ρ}{ε_{0}}$

Then,

$\nabla^{2} V_{3} = \frac{- ρ}{ε_{0}} + \frac{ρ}{ε_{0}} = 0$

As known to us,

$\nabla V_{3} = E_{3}$

$E_{3} = \nabla V_{3}$ as per derivation.

Determine proof of Greens identity

Substitute the above values in equation (1)

$\begin{array}{rcl} \int_{\nabla} [V_{3} (0) + E_{3}^{2}] d τ & = & - \oint V_{3} E_{3} \cdot d a \\ \int_{\nabla} E_{3}^{2} d τ & = & - \oint V_{3} E_{3} \cdot d a \end{array}$ …… (3)

As,

$E_{3} = E_{1} - E_{2}$ …… (4)

If V is specified as $V_{3} = 0$ or $E_{3} = 0$ then,

The equation (4) will be,

$0 = E_{1} - E_{2} E_{1} = E_{2}$

Thus, filed is uniquely determined.

And it is proved as this is a second uniqueness theorem.

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A more elegant proof of the second uniqueness theorem uses Green's
identity (Prob. 1.61c), with $T = U = V_{3}$ . Supply the details.

Short Answer

Step by step solution

Define function

Determine proof of Greens identity

Determine proof of Greens identity

One App. One Place for Learning.

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A more elegant proof of the second uniqueness theorem uses Green'sidentity (Prob. 1.61c), with T=U=V3. Supply the details.

Short Answer

Step by step solution

Define function

Determine proof of Greens identity

Determine proof of Greens identity

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Modern Physics

Circular Motion and Gravitation

Geometrical and Physical Optics

Famous Physicists

Further Mechanics and Thermal Physics

Rotational Dynamics

Study anywhere. Anytime. Across all devices.

A more elegant proof of the second uniqueness theorem uses Green's
identity (Prob. 1.61c), with $T = U = V_{3}$ . Supply the details.