Question

Consider a conductor of resistance ‘R’, length ‘L’, thickness‘d’ and resistivity ‘ρ’. Now this conductor is cut into four equal parts. What will be the new resistivity of each of these parts? Why? (ii) Find the resistance if all of these parts are connected in: (a) Parallel (b) Series (iii) Out of the combinations of resistors mentioned above in the previous part, for a given voltage which combination will consume more power and why?​

Answer 1

Answer:

i) Resistivity will remain same.

ii) a)$R_{equ}=\dfrac{R}{16}$

b)$R_{equ}=R$

iii)Parallel connection consume more power because it have low equivalent resistance.

Explanation:

Given that

Resistance =R

Length =L

Thickness=d

Resistivity =ρ

Now when wire cuts in 4 equal parts

L'= L/4

i)

We know that resistivity is a material properties and does not depends on the physical dimensions.So even wire cuts in 4 equal but still resistivity will remain same.

So the answer is, resistivity will be same.

ii)

We know that

R = ρ L/A

When L become 1/4 th then resistance for each part is R'

R' =  ρ L'/A  

L'= L/4

R' =  ρ L/4A  

R'= R/4

It means that resistance will become one forth of initial resistance

a)Parallel connection:

We know that equivalent resistance given as

$\dfrac{1}{R_{equ}}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}+\dfrac{1}{R_4}$

$R_1=R_2=R_3=R_4= \dfrac{R}{4}$

$\dfrac{1}{R_{equ}}=\dfrac{4}{R}+\dfrac{4}{R}+\dfrac{4}{R}+\dfrac{4}{R}$

$R_{equ}=\dfrac{R}{16}$

b)Series connection :

$R_{equ}=R_1+R_2+R_3+R_4$

$R_1=R_2=R_3=R_4= \dfrac{R}{4}$

$R_{equ}=R$

iii)

We know that Power P given as

P=V.I =V²/R                      ( V= IR)

For given voltage parallel connection consume more power because it have low equivalent resistance.

Answer 2

Answer:

Resistivity will remain same.

ii) a)R_{equ}=\dfrac{R}{16}R

equ

=

16

R

b)R_{equ}=RR

equ

=R

iii)Parallel connection consume more power because it have low equivalent resistance.

Explanation:

Given that

Resistance =R

Length =L

Thickness=d

Resistivity =ρ

Now when wire cuts in 4 equal parts

L'= L/4

i)

We know that resistivity is a material properties and does not depends on the physical dimensions.So even wire cuts in 4 equal but still resistivity will remain same.

So the answer is, resistivity will be same.

ii)

We know that

R = ρ L/A

When L become 1/4 th then resistance for each part is R'

R' = ρ L'/A

L'= L/4

R' = ρ L/4A

R'= R/4

It means that resistance will become one forth of initial resistance

a)Parallel connection:

We know that equivalent resistance given as

\dfrac{1}{R_{equ}}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}+\dfrac{1}{R_4}

R

equ

1

=

R

1

1

+

R

2

1

+

R

3

1

+

R

4

1

R_1=R_2=R_3=R_4= \dfrac{R}{4}R

1

=R

2

=R

3

=R

4

=

4

R

\dfrac{1}{R_{equ}}=\dfrac{4}{R}+\dfrac{4}{R}+\dfrac{4}{R}+\dfrac{4}{R}

R

equ

1

=

R

4

+

R

4

+

R

4

+

R

4

R_{equ}=\dfrac{R}{16}R

equ

=

16

R

b)Series connection :

R_{equ}=R_1+R_2+R_3+R_4R

equ

=R

1

+R

2

+R

3

+R

4

R_1=R_2=R_3=R_4= \dfrac{R}{4}R

1

=R

2

=R

3

=R

4

=

4

R

R_{equ}=RR

equ

=R

iii)

We know that Power P given as

P=V.I =V²/R ( V= IR)

For given voltage parallel connection consume more power because it have low equivalent resistance