Advertisements
Advertisements
प्रश्न
The electric current in a charging R−C circuit is given by i = i0 e−t/RC where i0, R and C are constant parameters of the circuit and t is time. Find the rate of change of current at (a) t = 0, (b) t = RC, (c) t = 10 RC.
Advertisements
उत्तर
Given: i = i0 e−t/RC
∴ Rate of change of current
\[= \frac{di}{dt}\]
\[= i_0 \left( \frac{- 1}{RC} \right) e^{- t/RC}\]
\[= \frac{- i_0}{RC} \times e^{- t/RC}\]
On applying the conditions given in the questions, we get:
(a) At \[t = 0, \frac{di}{dt} = \frac{- i_0}{RC} \times e^0 = \frac{- i_0}{RC}\]
(b) At \[t = RC, \frac{di}{dt} = \frac{- i_0}{RC} \times e^{- 1} = \frac{- i_0}{RCe}\]
(c) At \[t = 10 RC, \frac{dt}{di} = \frac{- i_0}{RC} \times e^{10}\] \[= \frac{- i_0}{RC e^{10}}\]
APPEARS IN
संबंधित प्रश्न
India has had a long and unbroken tradition of great scholarship — in mathematics, astronomy, linguistics, logic and ethics. Yet, in parallel with this, several superstitious and obscurantistic attitudes and practices flourished in our society and unfortunately continue even today — among many educated people too. How will you use your knowledge of science to develop strategies to counter these attitudes ?
If all the terms in an equation have same units, is it necessary that they have same dimensions? If all the terms in an equation have same dimensions, is it necessary that they have same units?
Suggest a way to measure the thickness of a sheet of paper.
\[\int\frac{dx}{\sqrt{2ax - x^2}} = a^n \sin^{- 1} \left[ \frac{x}{a} - 1 \right]\]
The value of n is
Find the dimensions of the coefficient of linear expansion α and
Test if the following equation is dimensionally correct:
\[h = \frac{2S cos\theta}{\text{ prg }},\]
where h = height, S = surface tension, ρ = density, I = moment of interia.
Test if the following equation is dimensionally correct:
\[V = \frac{\pi P r^4 t}{8 \eta l}\]
where v = frequency, P = pressure, η = coefficient of viscosity.
Test if the following equation is dimensionally correct:
\[v = \frac{1}{2 \pi}\sqrt{\frac{mgl}{I}};\]
where h = height, S = surface tension, \[\rho\] = density, P = pressure, V = volume, \[\eta =\] coefficient of viscosity, v = frequency and I = moment of interia.
Let x and a stand for distance. Is
\[\int\frac{dx}{\sqrt{a^2 - x^2}} = \frac{1}{a} \sin^{- 1} \frac{a}{x}\] dimensionally correct?
Can you have \[\vec{A} \times \vec{B} = \vec{A} \cdot \vec{B}\] with A ≠ 0 and B ≠ 0 ? What if one of the two vectors is zero?
Which of the sets given below may represent the magnitudes of three vectors adding to zero?
The resultant of \[\vec{A} \text { and } \vec{B}\] makes an angle α with \[\vec{A}\] and β with \[\vec{B}\],
The radius of a circle is stated as 2.12 cm. Its area should be written as
A carrom board (4 ft × 4 ft square) has the queen at the centre. The queen, hit by the striker moves to the from edge, rebounds and goes in the hole behind the striking line. Find the magnitude of displacement of the queen (a) from the centre to the front edge, (b) from the front edge to the hole and (c) from the centre to the hole.
Let \[\vec{a} = 2 \vec{i} + 3 \vec{j} + 4 \vec{k} \text { and } \vec{b} = 3 \vec{i} + 4 \vec{j} + 5 \vec{k}\] Find the angle between them.
If \[\vec{A} , \vec{B} , \vec{C}\] are mutually perpendicular, show that \[\vec{C} \times \left( \vec{A} \times \vec{B} \right) = 0\] Is the converse true?
Write the number of significant digits in (a) 1001, (b) 100.1, (c) 100.10, (d) 0.001001.
