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Zorluk: Çok zorElectric Circuits and Measuring Instruments

An electric cell of electromotive force EE and internal resistance rr is connected in series with a fixed resistor of resistance 8.0 Ω8.0\text{ }\Omega and a galvanometer of internal resistance 40.0 Ω40.0\text{ }\Omega. When a shunt resistor of resistance 10.0 Ω10.0\text{ }\Omega is connected in parallel across the galvanometer, the total current supplied by the cell increases by 50%50\%. What is the internal resistance rr of the cell in ohms?

Cevap: 48 \Omega

Cevap

The internal resistance of the cell is 48.0 Ω48.0\text{ }\Omega.
By determining the initial total circuit resistance RT1=48.0+rR_{T1} = 48.0 + r and the post-shunt total circuit resistance RT2=16.0+rR_{T2} = 16.0 + r, we apply Ohm's law with I2=1.5I1I_2 = 1.5 I_1. Equating total supply voltage EE gives E16.0+r=1.5×E48.0+r\frac{E}{16.0 + r} = 1.5 \times \frac{E}{48.0 + r}, which simplifies directly to r=48.0 Ωr = 48.0\text{ }\Omega.

Adım Adım Çözüm

1
Formulate the initial total resistance of the series circuit
RT1=8.0 Ω+40.0 Ω+r=48.0+rR_{T1} = 8.0\text{ }\Omega + 40.0\text{ }\Omega + r = 48.0 + r
Before the shunt is added, the fixed resistor, galvanometer, and internal resistance of the cell are all connected in series.
2
Calculate the effective resistance of the shunted galvanometer
Rp=40.0×10.040.0+10.0=8.0 ΩR_p = \frac{40.0 \times 10.0}{40.0 + 10.0} = 8.0\text{ }\Omega
Connecting the shunt in parallel across the galvanometer forms a parallel network.
3
Formulate the final total circuit resistance after shunting
RT2=8.0 Ω+8.0 Ω+r=16.0+rR_{T2} = 8.0\text{ }\Omega + 8.0\text{ }\Omega + r = 16.0 + r
The new circuit consists of the fixed resistor, the parallel shunted galvanometer combination, and the internal resistance in series.
4
Set up the ratio equation for total circuit currents based on the given 50%50\% current increase
E16.0+r=1.5(E48.0+r)\frac{E}{16.0 + r} = 1.5 \left(\frac{E}{48.0 + r}\right)
An increase of 50%50\% means I2=1.5I1I_2 = 1.5 I_1. According to Ohm's law, total current is inversely proportional to total circuit resistance for a constant EMF EE.
5
Solve the algebraic equation for internal resistance rr
r=48.0 Ωr = 48.0\text{ }\Omega
Cross-multiplying yields 48.0+r=24.0+1.5r48.0 + r = 24.0 + 1.5r, leading directly to 0.5r=24.00.5r = 24.0, so r=48.0 Ωr = 48.0\text{ }\Omega.

Anahtar Kavram

Internal Resistance and Circuit Modification via Galvanometer Shunting
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Electric Circuits and Measuring Instruments

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