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1

WB JEE 2022

MCQ (Single Correct Answer)
English
Bengali

Three concentric metallic shells A, B and C of radii a, b and c (a < b < c) have surface charge densities +$$\sigma$$, $$-$$$$\sigma$$ and +$$\sigma$$ respectively. The potential of shell B is

A
$$(a + b + c){\sigma \over {{\varepsilon _0}}}$$
B
$${{\sigma c} \over {{\varepsilon _0}}}$$
C
$$\left( {{{{a^2}} \over c} - {{{b^2}} \over c} + c} \right){\sigma \over {{\varepsilon _0}}}$$
D
$$\left( {{{{a^2}} \over c} - b + c} \right){\sigma \over {{\varepsilon _0}}}$$

Explanation

Net potential at any point $$(r = b)$$ of shell B

= potential due to shell A + potential due to shell B + potential due to shell C.

$$V = {V_{A,\,out}} + {V_{B,\,surface}} + {V_{C,\,in}}$$

$$ = {V_{A,\,out}} + {V_{B,\,surface}} + {V_{C,\,surface}}$$ (for shell, $${V_{in}} = {V_{surface}}$$)

$$V = {1 \over {4\pi {\varepsilon _0}}}\left[ {{{{q_a}} \over b} + {{{q_b}} \over b} + {{{q_c}} \over c}} \right]$$

$$ = {1 \over {4\pi {\varepsilon _0}}}\left[ {{{\sigma 4\pi {a^2}} \over b} + {{( - \sigma )4\pi {b^2}} \over b} + {{(\sigma )4\pi {c^2}} \over c}} \right]$$

$$ = {\sigma \over {{\varepsilon _0}}}\left[ {{{{a^2}} \over b} - b + c} \right]$$

a, b ও c (a < b < c) ব্যাসার্ধের তিনটি সমকেন্দ্রিক ধাতব খোলক A, B, C নেওয়া হল। A, B, C খোলক তিনটির আধানের তলমাত্রিক ঘনত্ব যথাক্রমে +$$\sigma$$, $$-$$$$\sigma$$, +$$\sigma$$ । B খলকের বিভব হবে

A
$$(a + b + c){\sigma \over {{\varepsilon _0}}}$$
B
$${{\sigma c} \over {{\varepsilon _0}}}$$
C
$$\left( {{{{a^2}} \over c} - {{{b^2}} \over c} + c} \right){\sigma \over {{\varepsilon _0}}}$$
D
$$\left( {{{{a^2}} \over c} - b + c} \right){\sigma \over {{\varepsilon _0}}}$$
2

WB JEE 2022

MCQ (Single Correct Answer)
English
Bengali

Find the equivalent capacitance between A and B of the following arrangement :

A
C
B
3C
C
$${{2C} \over 3}$$
D
$${{3C} \over 2}$$

Explanation

Here, all the three capacitors are connected in parallel. So, equivalent capacitance will be

$${C_{eq}} = C + C + C = 3C$$

চিত্রে প্রদর্শিত সমবায়টির A ও B এর মধ্যে তুল্য ধারকত্ব কত ?

A
C
B
3C
C
$${{2C} \over 3}$$
D
$${{3C} \over 2}$$
3

WB JEE 2022

MCQ (Single Correct Answer)
English
Bengali

A battery of emf E and internal resistance r is connected with an external resistance R as shown in the figure. The battery will act as a constant voltage source if

A
r << R
B
r >> R
C
r = R
D
It will never act as a constant voltage source.

Explanation

The constant voltage source provides a constant voltage to the load resistance regardless of variations or changes, in the load resistance. For this to happen, the source must have an internal resistance which is very low as compared to the resistance of the load.

i.e., $$r < < R$$

প্রদর্শিত চিত্রে, E তড়িচ্চালক বল ও r অভ্যন্তরীণ রোধের একটি ব্যাটারির সঙ্গে একটি বহিঃস্থ রোধ R যুক্ত করা হয়েছে। ব্যাটারিটি স্থির ভোল্টেজ উৎস হিসাবে কাজ করবে যদি

A
r << R
B
r >> R
C
r = R
D
কখনই স্থির ভোল্টেজ উৎস হিসাবে কাজ করবে না।
4

WB JEE 2021

MCQ (Single Correct Answer)
English
Bengali
Two metal wires of identical dimensions are connected in series. If $$\sigma$$1 and $$\sigma$$2 are the electrical conductivities of the metal wires respectively, the effective conductivity of the combination is
A
$$\sigma$$1 + $$\sigma$$2
B
$${{{\sigma _1}{\sigma _2}} \over {{\sigma _1} + {\sigma _2}}}$$
C
$${{2{\sigma _1}{\sigma _2}} \over {{\sigma _1} + {\sigma _2}}}$$
D
$${{{\sigma _1} + {\sigma _2}} \over {2{\sigma _1}{\sigma _2}}}$$

Explanation

Since, both wires are identical, so they have same cross-section A and same length l. Also, in series combination of wires, effective resistance,

R = R1 + R2

$$ \Rightarrow {{(l + l)} \over {{\sigma _{eff}}A}} = {1 \over {{\sigma _1}A}} + {1 \over {{\sigma _2}A}}$$

$$ \Rightarrow {2 \over {{\sigma _{eff}}}} = {1 \over {{\sigma _1}}} + {1 \over {{\sigma _2}}}$$

$$ \Rightarrow {\sigma _{eff}} = {{2{\sigma _1}{\sigma _2}} \over {{\sigma _1} + {\sigma _2}}}$$
একই আকারের দুটি ধাতব তারকে শ্রেনী সমবায়ে যুক্ত করা হল। যদি দুটি তারের তড়িৎ পরিবাহিতাঙ্ক $$\sigma$$1 এবং $$\sigma$$2 হয়, তবে এই সমবায়ের তুল্য পরিবাহিতাঙ্ক হবে
A
$$\sigma$$1 + $$\sigma$$2
B
$${{{\sigma _1}{\sigma _2}} \over {{\sigma _1} + {\sigma _2}}}$$
C
$${{2{\sigma _1}{\sigma _2}} \over {{\sigma _1} + {\sigma _2}}}$$
D
$${{{\sigma _1} + {\sigma _2}} \over {2{\sigma _1}{\sigma _2}}}$$

Explanation

যেহেতু উভয় তারের অভিন্ন, তাই তাদের একই প্রস্থচ্ছেদ A এবং একই দৈর্ঘ্য l এছাড়াও তারের সিরিজ সমন্বয়, কার্যকর প্রতিরোধের,

R = R1 + R2

$$ \Rightarrow {{(l + l)} \over {{\sigma _{eff}}A}} = {1 \over {{\sigma _1}A}} + {1 \over {{\sigma _2}A}}$$

$$ \Rightarrow {2 \over {{\sigma _{eff}}}} = {1 \over {{\sigma _1}}} + {1 \over {{\sigma _2}}}$$

$$ \Rightarrow {\sigma _{eff}} = {{2{\sigma _1}{\sigma _2}} \over {{\sigma _1} + {\sigma _2}}}$$

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