Find the angle between the planes whose vector equations are $$\vec { r } .\left( 2\hat { i } +2\hat { j } -3\hat { k } \right) =5$$ and $$\vec { r } .\left( 3\hat { i } -3\hat { j } +5\hat { k } \right) =3$$
The equations of the given planes are $$\vec { r } .\left( 2\hat { i }
+2\hat { j } -3\hat { k } \right) =5$$ and $$\vec { r } .\left( 3\hat {
i } -3\hat { j } +5\hat { k } \right) $$
It is known that if $$\vec { { n }_{ 1 } } $$ and $$\vec { { n }_{ 2 } } $$ are normal to the planes $$\vec { r } .\vec { { n }_{ 1 } } ={d}_{1}$$ and $$\vec { r }
.\vec { { n }_{ 2 } } ={d}_{2}$$ then the angle between them, $$\theta$$ is given by
$$\cos { \theta } =\left| \cfrac { \vec { { n }_{ 1 } } .\vec { {
n }_{ 2 } } }{ \left| \vec { { n }_{ 1 } } \right| \left| \vec { { n
}_{ 2 } } \right| } \right| ....(1)$$
Here $$\vec { { n }_{ 1 } }
=2\hat { i } +2\hat { j } -3\hat { k } $$ and $$\vec { { n }_{ 2 } }
=3\hat { i } -3\hat { j } +5\hat { k } $$
$$\therefore$$ $$\vec { { n
}_{ 1 } } .\vec { { n }_{ 2 } } =\left( 2\hat { i } +2\hat { j } -3\hat
{ k } \right) \left( 3\hat { i } -3\hat { j } +5\hat { k } \right)
=2.3+2.(-3)+(-3).5=-15$$
$$\left| \vec { { n }_{ 1 } } \right|
=\sqrt { { \left( 2 \right) }^{ 2 }+{ \left( 2 \right) }^{ 2 }+{
\left( -3 \right) }^{ 2 } } =\sqrt {17}$$
$$\left| \vec { { n }_{ 2 }
} \right| =\sqrt { { \left( 3 \right) }^{ 2 }+{ \left( -3 \right)
}^{ 2 }+{ \left( 5 \right) }^{ 2 } } =\sqrt {43}$$
Substituting the value of $$\vec { { n }_{ 1 } } .\vec { { n }_{ 2 } } $$ and $$\left|
\vec { { n }_{ 1 } } \right| \left| \vec { { n }_{ 2 } } \right| $$ in equation (1), we obtian
$$\cos { \theta } =\left| \cfrac { -15 }{ \sqrt { 17 } .\sqrt { 43 } } \right| $$
$$\Rightarrow$$ $$\cos { \theta } =\cfrac{15}{\sqrt {731}}$$
$$\Rightarrow \theta=$$ $$\cos ^{ -1 }\cfrac{15}{\sqrt {731}}$$
Let $$\Delta \,E$$ denote the energy gap between the valence band and the conduction band. The population of conduction electrons (and of the holes) is roughly proportional to $$e^{-\Delta \, E /2kT}$$. Find the ratio of the concentration of conduction electrons in diamond to that in silicon at a room temperature 300 K. $$\Delta \,E$$ for silicon is $$1\cdot1\, eV$$ and for diamond is $$6\cdot 0\, eV$$.
The conductivity of an intrinsic semiconductor depends on temperature as $$\sigma =\sigma _0 e^{-\Delta \, E/2kT}$$, where $$\sigma _0$$, is a constant. Find the temperature at which the conductivity of an intrinsic germanium semiconductor will be double of its value at T= 300 K. Assume that the gap for germanium is $$0\cdot 650\, eV$$ and remains constant as the temperature is increased.
In an intrinsic semiconductor the energy gap $$E_g$$ is $$1.2\ eV$$. Its hole mobility is much smaller than electron mobility and independent of temperature. What is the ratio between conductivity at $$600\ K$$ and that at $$300\ K$$? Assume that the temperature dependence of intrinsic carrier concentration $$n_i$$ is given by $$n_i = n_o exp \left(- \dfrac{E_g}{2 k_B T} \right)$$ Where $$n_0$$ is a constant.
The probability of electrons to be found in the conduction band of an intrinsic semiconductor at a finite temperature :-
In case of $$NPN$$ transistors the collector current is always less than the emitter current because
The introduced of a grid in a triode value affects plate current by
The mobility of free electron is greater than that of free holes because
The mobility of free electrons is greater than that of free holes because
Pure silicon crystal of length $$l (0.1m)$$ and area $$A$$($$10^{-4}m^2$$) has the mobility of electrons ($$\mu_e$$) and holes ($$\mu_h$$) as $$0.135 m^2/Vs$$ and $$0.048 m^2/Vs$$ , respectively. If the voltage applied across it is $$2V$$ and the intrinsic charge concentration is $$n_i = 1.5 \times 10^6 m^{-3}$$, then the total current flowing through the crystal is
The electric current in an extrinsic semi conductor is the same of currents due to holes and electrons.