प्रथम कोटि परिपथों (RC) के संबंध में निम्नलिखित कथनों पर विचार कीजिए:
1. RC चार्जिंग परिपथ में संधारित्र वोल्टेज घातांकीय रूप से बढ़ता है और आपूर्ति वोल्टेज तक अनंतस्पर्शी रूप से पहुँचता है।
2. RC परिपथ का समय स्थिरांक प्रतिरोध और धारिता के गुणनफल से दिया जाता है।
3. एक समय स्थिरांक के बाद संधारित्र वोल्टेज अपने अंतिम मान के लगभग 63 प्रतिशत तक पहुँचता है।
4. RC चार्जिंग परिपथ में धारा समय के साथ बढ़ती है और स्थिर अवस्था में अधिकतम होती है।
उपरोक्त में से कौन-सा/से कथन सही है/हैं?
उत्तर और चरणबद्ध हल
A — केवल 1, 2 और 3। v_C(t) = V(1 − e^(−t/RC)), τ = RC, t = τ पर ≈ 63.2%। चार्जिंग धारा i(t) = (V/R)e^(−t/RC) समय के साथ घटती है, बढ़ती नहीं।
विस्तृत हल (मूल भाषा में)
Statement-wise Verification
- Statement 1 - Correct.
In an RC charging circuit, capacitor voltage increases exponentially and approaches supply voltage asymptotically.
A first-order RC charging circuit responds exponentially because the capacitor voltage cannot change instantaneously. - Statement 2 - Correct.
Time constant of an RC circuit is given by product of resistance and capacitance.
The time constant τ=RC sets the speed - Statement 3 - Correct.
After one time constant, capacitor voltage reaches approximately 63 percent of its final value.
The time constant τ=RC sets the speed; at t=τ the capacitor reaches about 63.2% of its final voltage. - Statement 4 - Incorrect.
The current in RC charging circuit increases with time and reaches maximum at steady state.
Charging current starts at V/R and decays exponentially to zero
Core Concept
A first-order RC charging circuit responds exponentially because the capacitor voltage cannot change instantaneously. Starting from zero, capacitor voltage rises toward the source and charging current decays toward zero. The time constant τ=RC sets the speed; at t=τ the capacitor reaches about 63.2% of its final voltage.
Formula / Key Relationship
v_C(t)=V(1−e^(−t/RC)); i(t)=(V/R)e^(−t/RC); τ=RC.
Why the Other Options Are Wrong
Option B is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 2, 3. Option C is incorrect because it omits correct statement(s) 1. Option D is incorrect because it includes incorrect statement(s) 4.
Exam Shortcut / Approach
One time constant = 63.2%; five time constants ≈ settled.
Common Mistake
Charging current decreases with time; it does not increase toward steady state.
Quick Revision
RC Charging Response is linked with First-Order RC Circuit, Transient Analysis and Network Theory. One time constant = 63.2%; five time constants ≈ settled.
Consider the following statements regarding First Order Circuits (RC):
1. In an RC charging circuit, capacitor voltage increases exponentially and approaches supply voltage asymptotically.
2. Time constant of an RC circuit is given by product of resistance and capacitance.
3. After one time constant, capacitor voltage reaches approximately 63 percent of its final value.
4. The current in RC charging circuit increases with time and reaches maximum at steady state.
Which of the above statements is/are correct?
Answer & step-by-step solution
A first-order RC charging circuit responds exponentially because the capacitor voltage cannot change instantaneously. The time constant τ=RC sets the speed; at t=τ the capacitor reaches about 63.2% of its final voltage. Key relation: v_C(t)=V(1−e^(−t/RC)); i(t)=(V/R)e^(−t/RC); τ=RC.
Exam focus: One time constant = 63.2%; five time constants ≈ settled. Common trap: Charging current decreases with time; it does not increase toward steady state.
Statement-wise Verification
- Statement 1 - Correct.
In an RC charging circuit, capacitor voltage increases exponentially and approaches supply voltage asymptotically.
A first-order RC charging circuit responds exponentially because the capacitor voltage cannot change instantaneously. - Statement 2 - Correct.
Time constant of an RC circuit is given by product of resistance and capacitance.
The time constant τ=RC sets the speed - Statement 3 - Correct.
After one time constant, capacitor voltage reaches approximately 63 percent of its final value.
The time constant τ=RC sets the speed; at t=τ the capacitor reaches about 63.2% of its final voltage. - Statement 4 - Incorrect.
The current in RC charging circuit increases with time and reaches maximum at steady state.
Charging current starts at V/R and decays exponentially to zero
Core Concept
A first-order RC charging circuit responds exponentially because the capacitor voltage cannot change instantaneously. Starting from zero, capacitor voltage rises toward the source and charging current decays toward zero. The time constant τ=RC sets the speed; at t=τ the capacitor reaches about 63.2% of its final voltage.
Formula / Key Relationship
v_C(t)=V(1−e^(−t/RC)); i(t)=(V/R)e^(−t/RC); τ=RC.
Why the Other Options Are Wrong
Option B is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 2, 3. Option C is incorrect because it omits correct statement(s) 1. Option D is incorrect because it includes incorrect statement(s) 4.
Exam Shortcut / Approach
One time constant = 63.2%; five time constants ≈ settled.
Common Mistake
Charging current decreases with time; it does not increase toward steady state.
Quick Revision
RC Charging Response is linked with First-Order RC Circuit, Transient Analysis and Network Theory. One time constant = 63.2%; five time constants ≈ settled.
પ્રથમ ક્રમના પરિપથો (RC) અંગે નીચેનાં વિધાનો ધ્યાનમાં લો:
1. RC ચાર્જિંગ પરિપથમાં સંધારક વોલ્ટેજ ઘાતાંકીય રીતે વધે છે અને પુરવઠા વોલ્ટેજ તરફ અનંતસ્પર્શી રીતે પહોંચે છે.
2. RC પરિપથનો સમય અચળાંક અવરોધ અને ધારિતાના ગુણાકારથી મળે છે.
3. એક સમય અચળાંક પછી સંધારક વોલ્ટેજ તેના અંતિમ મૂલ્યના આશરે 63 ટકા સુધી પહોંચે છે.
4. RC ચાર્જિંગ પરિપથમાં પ્રવાહ સમય સાથે વધે છે અને સ્થિર અવસ્થામાં મહત્તમ થાય છે.
ઉપરોક્તમાંથી કયું/કયાં વિધાન સાચું/સાચાં છે?
જવાબ અને પગલાવાર ઉકેલ
A — ફક્ત 1, 2 અને 3. v_C(t) = V(1 − e^(−t/RC)), τ = RC, t = τ પર ≈ 63.2%. ચાર્જિંગ પ્રવાહ i(t) = (V/R)e^(−t/RC) સમય સાથે ઘટે છે, વધતો નથી.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Statement-wise Verification
- Statement 1 - Correct.
In an RC charging circuit, capacitor voltage increases exponentially and approaches supply voltage asymptotically.
A first-order RC charging circuit responds exponentially because the capacitor voltage cannot change instantaneously. - Statement 2 - Correct.
Time constant of an RC circuit is given by product of resistance and capacitance.
The time constant τ=RC sets the speed - Statement 3 - Correct.
After one time constant, capacitor voltage reaches approximately 63 percent of its final value.
The time constant τ=RC sets the speed; at t=τ the capacitor reaches about 63.2% of its final voltage. - Statement 4 - Incorrect.
The current in RC charging circuit increases with time and reaches maximum at steady state.
Charging current starts at V/R and decays exponentially to zero
Core Concept
A first-order RC charging circuit responds exponentially because the capacitor voltage cannot change instantaneously. Starting from zero, capacitor voltage rises toward the source and charging current decays toward zero. The time constant τ=RC sets the speed; at t=τ the capacitor reaches about 63.2% of its final voltage.
Formula / Key Relationship
v_C(t)=V(1−e^(−t/RC)); i(t)=(V/R)e^(−t/RC); τ=RC.
Why the Other Options Are Wrong
Option B is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 2, 3. Option C is incorrect because it omits correct statement(s) 1. Option D is incorrect because it includes incorrect statement(s) 4.
Exam Shortcut / Approach
One time constant = 63.2%; five time constants ≈ settled.
Common Mistake
Charging current decreases with time; it does not increase toward steady state.
Quick Revision
RC Charging Response is linked with First-Order RC Circuit, Transient Analysis and Network Theory. One time constant = 63.2%; five time constants ≈ settled.
More Transient Analysis questions