1
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 9th January Evening Slot

A parallel plate capacitor with square plates is filled with four dielecytrics of dielectrics constants K1, K2, K3, K4 arranged as shown in the figure. The effective dielectric constant K will be :

A
$$K = {{{K_1}{K_2}} \over {{K_1} + {K_2}}} + {{{K_3}{K_4}} \over {{K_3} + {K_4}}}$$
B
$$K = {{\left( {{K_1} + {K_2}} \right)\left( {{K_3} + {K_4}} \right)} \over {2\left( {{K_1} + {K_2} + {K_3} + {K_4}} \right)}}$$
C
$$K = {{\left( {{K_1} + {K_2}} \right)\left( {{K_3} + {K_4}} \right)} \over {{K_1} + {K_2} + {K_3} + {K_4}}}$$
D
$$K = {{\left( {{K_1} + {K_4}} \right)\left( {{K_2} + {K_3}} \right)} \over {2\left( {{K_1} + {K_2} + {K_3} + {K_4}} \right)}}$$

Explanation


Here, $${C_1} = {{{K_1}{\varepsilon _0}.L.{L \over 2}} \over {{d \over 2}}} = {{{K_1}{\varepsilon _0}{L^2}} \over d}$$

$${C_2}{{{K_2}{\varepsilon _0}.L.{L \over 2}} \over {{d \over 2}}} = {{{K_2}{\varepsilon _0}{L^2}} \over d}$$

$${C_3} = {{{K_3}{\varepsilon _0}{L^2}} \over d}$$

$${C_4} = {{{K_4}{\varepsilon _0}{L^2}} \over d}$$

Here  C1 C2 are in series and C3, C4 are in series.

Equivalent capacitance point A and B is,

Ceq $$=$$ $${{{C_1}{C_2}} \over {{C_1} + {C_2}}} + {{{C_3}{C_4}} \over {{C_3} + {C_4}}}$$

$$ = {C_{12}} + {C_{34}}$$

$${C_{12}}\,\, = \,\,{{{{{K_1}{\varepsilon _0}{L^2}} \over d} \times {{{K_2}{\varepsilon _0}{L^2}} \over d}} \over {{{{K_1}{\varepsilon _0}{L^2}} \over d} + {{{K_2}{\varepsilon _0}{L^2}} \over d}}}$$

$$ = \,\,{{{K_1}{K_2}} \over {{K_1} + {K_2}}} \times {{{\varepsilon _0}{L^2}} \over d}$$

Similarly,

$${C_{34}} = \,\,{{{K_3}{K_4}} \over {{K_3} + {K_4}}} \times {{{\varepsilon _0}{L^2}} \over d}$$

$$ \therefore $$  Ceq $$=$$ $$\left[ {{{{K_1}{K_2}} \over {{K_1} + {K_2}}} + {{{K_3}{K_4}} \over {{K_3} + {K_4}}}} \right]{{{\varepsilon _0}{L^2}} \over d}$$

$$ \Rightarrow $$  $${{K{\varepsilon _0}{L^2}} \over d}$$  $$ = \left[ {{{{K_1}{K_2}} \over {{K_1} + {K_2}}} + {{{K_3}{K_4}} \over {{K_3} + {K_4}}}} \right]{{{\varepsilon _0}{L^2}} \over d}$$

$$ \Rightarrow $$  $$K = {{{K_1}{K_2}} \over {{K_1} + {K_2}}} + {{{K_3}{K_4}} \over {{K_3} + {K_4}}}$$
2
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 10th January Morning Slot

A parallel plate capacitor is of area 6 cm2 and a separation 3 mm. The gap is filled with three dielectric materials of equal thickness (see figure) with dielectric constants K1 = 10, K2 = 12 and K3 = 14. The dielectric constant of a material which when fully inserted in above capacitor, gives same capacitance would be -

A
12
B
36
C
14
D
4

Explanation

Let dielectric constant of material used be K.

$$ \therefore $$  $${{10{ \in _0}\,A/3} \over d} + {{12{ \in _0}\,A/3} \over d} + {{14{ \in _0}\,A/3} \over d} = {{K{ \in _0}A} \over d}$$

$$ \Rightarrow $$    K $$=$$ 12
3
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 10th January Morning Slot

A uniform metallic wire has a resistance of 18 $$\Omega $$ and is bent into an equilateral triangle. Then, the resistance between any two vertices of the triangle is -
A
12 $$\Omega $$
B
2 $$\Omega $$
C
4 $$\Omega $$
D
8 $$\Omega $$

Answer

$$\Omega $$

Explanation



Req berween any two vertex will be

$${1 \over {{{\mathop{\rm R}\nolimits} _{eq}}}} = {1 \over {12}} + {1 \over 6} \Rightarrow {{\mathop{\rm R}\nolimits} _{eq.}} = 4\Omega $$
4
MCQ (Single Correct Answer)

JEE Main 2019 (Online) 10th January Morning Slot

A 2 W carbon resistor is color coded with green, black, red and brown respectively. The maximum current which can be passed through this resistor is -
A
0.4 mA
B
20 mA
C
63 mA
D
100 mA

Explanation

P = i2R.

$$ \therefore $$   for imax, R must be minimum

from color coding R = 50 $$ \times $$ 102$$\Omega $$

$$ \therefore $$   imax = 20mA

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