K A Tsokos Solutions for Chapter: Fields (HL), Exercise 2: Test yourself

Two stars of equal masses $\mathrm{M}$ orbit a common mass as shown in the diagram. The radius of orbit of each star is $\mathrm{R}$. Assume that each star has a mass equal to $1.5 \mathrm{solar} \mathrm{mass}$  (solar mass=$2.0\times {10}^{30}\mathrm{kg}$ ) and the initial separation of the star is $2\times {10}^{9 }\mathrm{m}$.

. The orbital period decreases at a rate of $\frac{{\displaystyle \raisebox{1ex}{$△\mathrm{T}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{T}$}\right.}}{△\mathrm{t}}=72 {\mathrm{\mu syr}}^{-1}.$  The two stars will collapse into each other when $△\mathrm{E}\simeq \mathrm{E}$. Estimate the lifetime, in years of this binary star system

A charge $-\mathrm{q}$ whose mass $\mathrm{m}$  moves in a circle of radius $\mathrm{r}$ around another positive charge $\mathrm{q}$ located at the centre of the circle, as shown in the diagram.

 Draw the force on the moving charge.

A charge $-\mathrm{q}$ whose mass $\mathrm{m}$  moves in a circle of radius $\mathrm{r}$ around another positive charge $\mathrm{q}$ located at the centre of the circle, as shown in the diagram.

Show that the velocity of the charge is given by ${\mathrm{v}}^2=\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{{\mathrm{q}}^2}{\mathrm{mr}}$

A charge $-\mathrm{q}$ whose mass $\mathrm{m}$  moves in a circle of radius $\mathrm{r}$ around another positive charge $\mathrm{q}$ located at the centre of the circle, as shown in the diagram.

Show that the total energy of the charge is given by $\mathrm{E}=-\frac{1}{8{\mathrm{\pi \epsilon }}_0}\frac{{\mathrm{q}}^2}{\mathrm{r}}$

A charge $-\mathrm{q}$ whose mass $\mathrm{m}$  moves in a circle of radius $\mathrm{r}$ around another positive charge $\mathrm{q}$ located at the centre of the circle, as shown in the diagram.

Show that the total energy of the charge is given by $\mathrm{E}=-\frac{1}{8{\mathrm{\pi \epsilon }}_0}\frac{{\mathrm{q}}^2}{\mathrm{r}}$. Hence determine how much energy must be supplied to the charge if it is to be orbit around the stationary charge at a radius equal to $2\mathrm{r}$.

An electron of charge $-\mathrm{e}$ and mass $\mathrm{m}$ orbit the proton in a hydrogen atom. Show that the period of revolution of the electron is given by 

${\mathrm{T}}^2=\frac{4{\mathrm{\pi }}^2 \mathrm{m}}{\mathrm{k} {\mathrm{e}}^2}{\mathrm{r}}^3$  where $\mathrm{k}$= Coulombs' constant. and  $r$= radius of the orbit.

An electron of charge $-\mathrm{e}$ and mass $\mathrm{m}$ orbit the proton in a hydrogen atom. Show that the period of revolution of the electron is given by 

${\mathrm{T}}^2=\frac{4{\mathrm{\pi }}^2 \mathrm{m}}{\mathrm{k} {\mathrm{e}}^2}{\mathrm{r}}^3$  where $\mathrm{k}$= Coulombs' constant. and  $r$= radius of the orbit.

Calculate this period for an orbit radius of $0.5\times {10}^{-10}\mathrm{m}$

An electron of charge $-\mathrm{e}$ and mass $\mathrm{m}$ orbit the proton in a hydrogen atom. Show that the period of revolution of the electron is given by 

${\mathrm{T}}^2=\frac{4{\mathrm{\pi }}^2 \mathrm{m}}{\mathrm{k} {\mathrm{e}}^2}{\mathrm{r}}^3$  where $\mathrm{k}$= Coulombs' constant. and  $r$= radius of the orbit.

Using the results of the previous problem, calculate the energy that must be supplied to the electron so it orbits the proton in an orbit of 

$2.0\times {10}^{-10}\mathrm{m}$