Question

In Fig. 27-19, a circuit consists of a battery and two uniform resistors, and the section lying along an xaxis is divided into five segments of equal lengths.

(a) Assume that${\mathbf{R}}_1\mathbf{=}{\mathbf{R}}_2$and rank the segments according to the magnitude of the average electric field in them, greatest first.

(b) Now assume that${\mathbf{R}}_1\mathbf{>}{\mathbf{R}}_2$and then again rank the segments.

(c) What is the direction of the electric field along the xaxis?

Short answer

1. Rank of electric field for $R_1=R_2$is $b=d>a=c=e$
2. Rank of electric field for $R_1>R_2$is$b>d>a=c=e$
3. Direction of electric field is along positive x direction.

Step-by-step solution

Step 1: Given

$$\begin{gathered} R_1=R_2 \\ {R}_1>R_2 \end{gathered}$$

Determining the concept

Here, Use the equation of Ohm’s law and the relation between the voltage and the electric field together and make the equation for the electric field relating with the resistance and the length of segment.

Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points.

Formulae are as follow:

$$\begin{gathered} V=IR \\ V=EL \end{gathered}$$

Where, I is current, V is voltage, R is resistance, E is electric field.

(a) Determining the rank of segments according to the magnitude of the electric field greatest first forR1=R2

According to Ohm’s law,

$V=IR$

And relation between voltage and electric field is as follow:

$V=EL$

So,

$EL=IR$

$E=\frac{IR}{L}$

Here, the resistance through the wire is less than through the circuits and length of each segment is same.

So, the electric field only depends on R,

Segments a, c and e have same resistance.

So, they have the same electric field. Also, the resistance$R_1=R_2$are greater than the resistance through the segments, So, the electric field through b and d is the same but greater than segments a, c and e.

Hence, the rank will become $b=d>a=c=e$

(b) Determining the rank of segments according to the magnitude of the electric field greatest first for R1>R2

Here, the resistance through the wire is less than through the circuits and the length of each segment is same.

So, the electric field only depends on R,

Segments a, c and e have the same resistance.

So, they have same electric field,

And resistance $R_1>R_2$and so, The electric field through $b>d$

$b>d>a=c=e$

Hence,rank of electric field for $R_1>R_2$is$b>d>a=c=e$

(c) Determining the direction of electric field along the x axis

The direction of the electric field is from more positive end to more negative end in the circuit. So, it is in positive x direction.

Hence, direction of electric field is along positive x direction.

Therefore, use Ohm’s law to find the voltage and the relation between the voltage and the electric field to find the electric field.