||Message]
An inductor, made up of a very long air-core solenoid of length $\ell$, radius $r$ and $n$ turns per unit length, carries a current $I$. For $\frac{dI}{dt}>0$, verify that \begin{equation} \frac{dU_B}{dt} = -\iint_S \vec{S}.\overrightarrow{dA}, \end{equation} where the surface $S$ in the surface integral in the right hand side is the surface that encloses the solenoid and \(U_B\) is the energy density associated with the magentic field.
Analysis
You need to understand the following. Read the given question several times and answer the following questions. After making sure that you have correct answers and that you understand them as well, try to solve the problem.
- What is Poynting theorem?
- What does $\vec{S}$ in \EqRef{EQ02} stand for?
- What is the expression of the Poynting vector?
- What physical quantities do the symbols $U_B$ and $\vec{S}$, in \EqRef{EQ02}, represent?
- Is it correct that the expressions for the quantities $\vec{S}$ and $U_B$ involve $\vec{E}$ and $\vec{B}$? If not, what is the correct expression?
- What field(s) does the solenoid produce? Magnetic field $\vec{B}$? Electric field $\vec{E}$? Both of these? None of these? Do you know the expressions?
- Compute the magnetic field produced by the solenoid. Write zero as your answer if it does not produce it.
- Repeat the above question for the electric field.
- Find $\vec{S}$ in terms of current $I$ and compute the surface integral $\iint_S \vec{S}.\overrightarrow{dA}$. Find $\vec{S}$ on the sutface of the solenoid.
- What will be the volume integral of $U_B$?
Exclude node summary :
n
Exclude node links:
0
4934:EM-HOME, 4727: DIAMOND POINT
0
