Capacitors and capacitance
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Define capacitance.
Capacitance (C) is the charge (Q) stored per unit potential difference (V) across a capacitor. Measured in Farads (F), where 1 F = 1 C/V.
State the formula that relates capacitance, charge, and potential difference.
The relationship between capacitance (C), charge (Q), and potential difference (V) is given by the formula: C = Q / V.
Derive the formula for the total capacitance of capacitors connected in parallel.
In parallel, the potential difference across each capacitor is the same. The total charge stored is the sum of charges on each capacitor (Q_total = Q1 + Q2 + ...). Thus, C_total = C1 + C2 + ...
Derive the formula for the total capacitance of capacitors connected in series.
In series, the charge on each capacitor is the same. The total potential difference is the sum of potential differences across each capacitor (V_total = V1 + V2 + ...). Thus, 1/C_total = 1/C1 + 1/C2 + ...
A 5μF capacitor is charged to 10V. What charge does it store?
Using C = Q/V, we have Q = CV = (5 × 10⁻⁶ F)(10 V) = 5 × 10⁻⁵ C or 50 μC.
Two capacitors, 2μF and 4μF, are connected in series. What is the total capacitance?
Using 1/C_total = 1/C1 + 1/C2, we have 1/C_total = 1/(2×10⁻⁶) + 1/(4×10⁻⁶). Therefore, C_total = 1.33 μF.
Two capacitors, 3μF and 6μF, are connected in parallel. What is the total capacitance?
Using C_total = C1 + C2, we have C_total = (3 × 10⁻⁶ F) + (6 × 10⁻⁶ F) = 9 × 10⁻⁶ F or 9 μF.
Describe how an isolated spherical conductor stores charge and relates to its capacitance.
An isolated spherical conductor stores charge uniformly on its surface. The capacitance is proportional to its radius; a larger sphere can store more charge at a given potential. C = 4πε₀r, where r is the radius.
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