*Notes to a video lecture on http://www.unizor.com*

__Direct Current - Ohm's Law -__

Problems 2

Problems 2

*Problem A*

Two resistors

*and*

**R**_{1}*are connected parallel to each other in a circuit.*

**R**_{2}Prove that the currents flowing through them are inversely proportional to their resistances, that is

*.*

**I**_{1}/I_{2}=R_{2}/R_{1}*Hint*

It follows directly from the Ohm's Law.

*Problem B*

Given a square

*with diagonal*

**ABCD***.*

**AC**Each side of this square and diagonal

*are resistors with resistance*

**AC***each.*

**R**Points

*and*

**A***are connected to a source of electricity with voltage*

**D***.*

**U**Determine resistance of an entire circuit

*and current*

**R**_{ABCD}*along the diagonal*

**I**_{AC}*.*

**AC***Solution*

Redraw the circuit as

and start with calculating the resistance of elements in a thin rectangle.

**1/R**_{ABC}= 1/R + 1/(2R) = 3/(2R)

**R**_{ABC}= 2R/3Now the total resistance of a circuit is calculated as a parallel connection of the resitors with resistance

*at the top branch and another*

**(2R/3)+R=5R/3***at the bottom of a drawing*

**R**

**1/R**

= 3/(5R) + 1/R = 8/(5R)_{ABCD}= [1/(5R/3)] + 1/R == 3/(5R) + 1/R = 8/(5R)

Total resistance of a circuit is

**R**_{ABCD}= 5R/8Current from the source of electricity in the common part of a circuit is, therefore,

**I = 8U/(5R)**To find the current along diagonal

*, we need to know the voltage between points*

**AC***and*

**A***.*

**C**The voltage between points

*and*

**A***is*

**D***.*

**U**The current in a common wire to the top branch with resistance

*is, therefore,*

**5R/3**

**I**_{top}= 3U/(5R)The voltage in that branch drops from point

*to point*

**A***by*

**C**

**U**

= 3U/(5R) · 2R/3 = 2U/5_{ABC}= I_{top}·R_{ABC}== 3U/(5R) · 2R/3 = 2U/5

Therefore, the current in the diagonal

*is*

**AC**

**I**_{AC}=

**R**_{ABCD}= 5R/8

**I**_{AC}= 2U/(5R)*Problem C*

Given a cube

*made with identical resistors*

**ABCDEFGH***on each edge, 12 resistors altogether.*

**R**It is connected to the source of electricity at two vertices on its main diagonal

*.*

**AG**What is the total resistance

*of this cube?*

**R**_{cube}*Solution*

The cube is a symmetrical figure relatively to its main diagonal

*and, since all resistors are the same, one on each edge, we can use this consideration to state that the main current from the source of electricity to point*

**AG****is divided into three identical currents along edges**

*A**,*

**AB***and*

**AD***.*

**AE**Therefore, a voltage drop at points

*,*

**B***and*

**D***is exactly the same and there is no difference in electric potential between these three vertices.*

**E**If there is no difference in electric potential between points

*,*

**B***and*

**D***, there will be no movement of electrons between these three points if we merge these three points into one point*

**E***. The flow of electricity will not change by this merging. Analogously, the main current from the source of electricity to point*

**X****is divided into three identical currents along edges**

*G**,*

**GC***and*

**GF***.*

**GH**Therefore, a voltage drop at points

*,*

**C***and*

**F***is exactly the same and there is no difference in electric potential between these three vertices.*

**H**If there is no difference in electric potential between points

*,*

**C***and*

**F***, there will be no movement of electrons between these three points if we merge these three points into one point*

**H***. The flow of electricity will not change by this merging.*

**Y**The following picture is the result of this transformation.

Regardless of seemingly complex picture, it actually represents a simple circuit.

Between points

*and*

**A***there are three identical parallel resistors. Their combined resistance is*

**X***.*

**R**_{AX}

**R**_{AX}= R/3Between points

*and*

**X***there are six identical parallel resistors. Their combined resistance is*

**Y***.*

**R**_{XY}

**R**_{XY}= R/6Between points

*and*

**Y***there are three identical parallel resistors. Their combined resistance is*

**G***.*

**R**_{YG}

**R**_{YG}= R/3These three groups are connected in a series with combined resistance

**R**_{cube}= R/3 + R/6 R/3 = 5R/6
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