Electrodynamics * – Page 3

February 19, 2019January 17, 2023

201902190351 Electric circuit diagrams (Elementary) Q10

The blogger claims no originality of his question below.

Find the equivalent resistance between nodes $e$ and $f$ .

Circuit0009

Solution:

We notice that the leftmost resistor is short-circuited, and we have the following diagram:

Circuit0009Sol001

Or, more neatly drawn, we have

Circuit0009Sol002

The equivalent resistance is $R_{\mathrm{eq}}=R+\displaystyle{\frac{R\times R}{R+R}}+R=2.5R$ .

February 19, 2019January 17, 2023

201902190342 Electric circuit diagrams (Elementary) Q9

The blogger claims no originality of his question below.

Find the equivalent resistance between nodes $a$ and $b$ .

Circuit0007

Solution.

The equivalent resistance of two $R$ ‘s in series is $2R$ :

Circuit0007Sol001

The equivalent resistance of $R$ and $2R$ in parallel is $R_{\mathrm{eq}}=\displaystyle{\frac{(R)(2R)}{R+2R}}=\displaystyle{\frac{2}{3}R}$ :

Circuit0007Sol002

For $R$ and $\displaystyle{\frac{2R}{3}}$ in series, $R_{\mathrm{eq}}=\displaystyle{\frac{5}{3}R}$ :

Circuit0007Sol003

Finally, the equivalent resistance between nodes $a$ and $b$ is $R_{\mathrm{eq}}=\displaystyle{\frac{R\bigg( \displaystyle{\frac{5}{3}R}\bigg)}{R+\displaystyle{\frac{5}{3}R}}}=\displaystyle{\frac{5}{8}R}$ .

February 19, 2019January 17, 2023

201902190332 Electric circuit diagrams (Elementary) Q8

The blogger claims no originality of his problem below.

Find the equivalent resistance between nodes $p$ and $q$ .

Circuit0008

Solution.

We draw the following diagram:

Circuit0008Sol001

Then the equivalent resistance is $R_{\mathrm{eq}}=R+\bigg( \displaystyle{\frac{1}{R}+\frac{1}{R}+\frac{1}{R}}\bigg)^{-1}+R=\displaystyle{\frac{8}{3}R}$ .

February 19, 2019January 17, 2023

201902190326 Electric circuit diagrams (Elementary) Q7

The blogger claims no originality of his problem below.

Circuit0006

Find the equivalent resistance of the circuit above.

Solution.

We draw the diagram below:

Circuit0006Sol001

For two resistors $R$ in parallel:

$R_{\mathrm{eq}}=\displaystyle{\frac{R}{2}}$ .

For three resistors $R$ in parallel:

$R_{\mathrm{eq}}=\displaystyle{\frac{R}{3}}$ .

Summing over them in series:

$R_{\mathrm{eq}}=\displaystyle{\frac{R}{2}+\frac{R}{3}+\frac{R}{2}}=\displaystyle{\frac{4}{3}R}$ .

February 19, 2019October 18, 2023

201902190310 Electric circuit diagrams (Elementary) Q6

The blogger claims no originality of his question below.

Circuit0002

In the figure above, the resistors are identical. Find the equivalent resistance between nodes:

$a$ and $b$ ;
$c$ and $d$ ;
$b$ and $c$ .

Solution.

We label the nodes by $A$ , $B$ , $C$ , etc. Different nodes with the same electric potential will be labelled with the same label, as shown below:

Assuming resistors are identical, there is no current flowing through the central resistor between two equipotential end-nodes $B$ ‘s. Hence we draw the diagram below:

The equivalent resistance is $R_{\mathrm{eq}}=\displaystyle{\bigg( \frac{1}{R+R}+\frac{1}{R+R}\bigg)^{-1}}=R$ .(Alternatively one can use the $Y$ – $\Delta$ transformation.)
We draw the diagram below:

The equivalent resistance between nodes $c$ and $d$ is given by $\displaystyle{\frac{1}{R_{\mathrm{eq}}}}=\displaystyle{\frac{1}{R+R}}+\displaystyle{\frac{1}{R}}+\displaystyle{\frac{1}{R+R}}$ . Thus $R_{\mathrm{eq}}=\displaystyle{\frac{R}{2}}$ .
By $Y$ – $\Delta$ transformation.

February 19, 2019January 17, 2023

201902190246 Electric circuit diagrams (Elementary) Q5

The blogger claims no originality of his problem below.

Circuit0012

Find the equivalent resistance of the circuit above.

Solution.

In series, $5\,\mathrm{\Omega} +9\,\mathrm{\Omega} +1\,\mathrm{\Omega} =15\,\mathrm{\Omega}$ .

Circuit0012Sol001

In parallel, $\bigg(\displaystyle{\frac{1}{10\,\mathrm{\Omega} }+\frac{1}{15\,\mathrm{\Omega} }}\bigg)^{-1}=6\,\mathrm{\Omega}$ .

Circuit0012Sol002

In series, $4\,\mathrm{\Omega} +6\,\mathrm{\Omega} +2\,\mathrm{\Omega} =12\,\mathrm{\Omega}$ .

Circuit0012Sol003

In parallel, $\bigg(\displaystyle{\frac{1}{4\,\mathrm{\Omega} }+\frac{1}{12\,\mathrm{\Omega} }}\bigg)^{-1}=3\,\mathrm{\Omega}$ .

Circuit0012Sol004

Finally, the equivalent resistance of the resistors above in series is $R_{\mathrm{eq}}=3\,\mathrm{\Omega} +3\,\mathrm{\Omega} +4\,\mathrm{\Omega} =10\,\mathrm{\Omega}$ .

February 18, 2019January 17, 2023

201902182224 Electric circuit diagrams (Elementary) Q4

The blogger claims no originality of his question below.

Circuit0004

Find the equivalent resistance, in terms of $R_1$ , $R_2$ , $R_3$ , and $R_4$ , when the switch is

i. open;

ii. closed.

Solution.

i. When the switch is open, we draw the following diagram: Circuit0004Sol001 The equivalent resistance of $R_1$ and $R_3$ in series is their sum $R_1+R_3$ . Similarly, that of $R_2$ and $R_4$ in series is $R_2+R_4$ . The equivalent resistance $R_{\mathrm{eq}}$ of $R_1+R_3$ and $R_2+R_4$ in parallel is given by $\displaystyle{\frac{1}{R_{\mathrm{eq}}}=\frac{1}{R_1+R_3}+\frac{1}{R_2+R_4}}$ . Thus $R_{\mathrm{eq}}=\displaystyle{\frac{(R_1+R_3)(R_2+R_4)}{R_1+R_2+R_3+R_4}}$ .

ii. When the switch is closed, we draw the following diagram: Circuit0004Sol002 The equivalent resistance of $R_1$ and $R_2$ in parallel is $\displaystyle{\frac{R_1R_2}{R_1+R_2}}$ . Similarly, that of $R_3$ and $R_4$ in parallel is $\displaystyle{\frac{R_3R_4}{R_3+R_4}}$ . Thus the equivalent resistance of them in series is $R_{\mathrm{eq}}=\displaystyle{\frac{R_1R_2}{R_1+R_2}+\frac{R_3R_4}{R_3+R_4}}$ .

February 18, 2019January 17, 2023

201902180835 Electric circuit diagrams (Elementary) Q3

The blogger claims no originality of his question below.

0004

In the figure above, when the key is closed, which of the following descriptions about the circuit is true?

1. If $A$ is an ammeter and $B$ is a voltmeter, then resistors $R_1$ and $R_2$ are in parallel.
2. If $A$ is an ammeter and $B$ is a voltmeter, then resistors $R_1$ and $R_2$ are in series.
3. If $A$ is a voltmeter and $B$ is an ammeter, then resistors $R_1$ and $R_2$ are in parallel.
4. If $A$ is a voltmeter and $B$ is an ammeter, then resistors $R_1$ and $R_2$ are in series.

Answer. A

February 18, 2019January 17, 2023

201902180824 Electric circuit diagrams (Elementary) Q2

The blogger claims no originality of his question below.

0002

In the figure above, the same current pass through the same cuboid resistor with sides $a$ , $b$ and $c$ where $a>b>c$ . The resistor has a uniform composition. The current direction is indicated by the arrow. In which direction (A, B or C) does the current have the least resistance?

Answer. A

February 18, 2019January 17, 2023

201902180817 Electric circuit diagrams (Elementary) Q1

The blogger claims no originality of his question below.

0001

In the figure above, resistors $R_1$ , $R_2$ , $R_3$ and $R_4$ are connected in series. It is known that $R_3=R_4$ . After the circuit is connected to a battery, the voltmeter readings of $V_1$ and $V_2$ are $10\,\mathrm{V}$ and $5\,\mathrm{V}$ respectively. Find the potential difference (p.d.) across points $a$ and $b$ .

Answer. $15\,\mathrm{V}$ is an answer.