Current through bulb 4 is maximum

To determine the resistance of a carbon resistor with color bands brown, black, green, and gold, use the color code for resistors:

1. Color code values:

• Brown:
  $1$ 

  (1st digit)

• Black:
  $0$ 

  (2nd digit)

• Green:
  $10^5$ 

  (multiplier)

• Gold:
  $\pm 5\%$ 

  (tolerance)

2. Calculate resistance:

The resistance is calculated as:

$$R = (\text{1st digit}\,\times\,10 + \text{2nd digit}) \,\times\, \text{multiplier}.$$

Substitute values:

$$R = (1 \times 10 + 0) \times 10^5 = 10 \times 10^5 = 10^6 \, \Omega.$$

This is equal to:

$$R = 1 \, \text{M}\Omega \, (\text{megaohm}).$$

3. Tolerance:

The gold band indicates a tolerance of

$\pm 5\%$

, so the resistance can vary between:

$$1 \, \text{M}\Omega \pm 5\%.$$

Final Answer:

The resistance is:

$$\boxed{1 \, \text{M}\Omega \, \pm 5\%.}$$

When

$N$

identical cells, each of emf

$e$

and internal resistance

$r$

, are connected in series and

$n$

cells are connected in reverse, the resulting emf and internal resistance of the battery can be determined as follows:

1. Resultant emf ( $\varepsilon_0$

):

• For correctly connected cells, the total emf is:
  $$\varepsilon_{\text{correct}} = (N - n)e.$$ 
• For n reversed cells, their emf opposes the total emf. The opposing emf is:
  $$\varepsilon_{\text{reverse}} = ne.$$ 

  The net emf of the resulting battery is: 

  $$\varepsilon_0 = \varepsilon_{\text{correct}} - \varepsilon_{\text{reverse}} = (N - n)e - ne = (N - 2n)e.$$ 

2. Resultant internal resistance ( $r_0$

):

• All cells, whether correctly or incorrectly connected, contribute to the total internal resistance because resistances add in series. The total internal resistance is:
  $$r_0 = N r.$$ 

Final Answer:

The emf and internal resistance of the resulting battery are:

$$\boxed{\varepsilon_0 = (N - 2n)e, \quad r_0 = Nr.}$$

To find the temperature coefficient of resistance

$\alpha$

of the conductor, we use the formula for the change in resistance with temperature:

$$R_T = R_0 \left( 1 + \alpha T \right)$$

Where:

• 
  $R_T$ 

  is the resistance at temperature $T$ 

  ,

• 
  $R_0$ 

  is the resistance at the reference temperature (0°C),

• 
  $\alpha$ 

  is the temperature coefficient of resistance,

• 
  $T$ 

  is the temperature change in °C.

Step 1: Given

• The ratio of resistances at 15°C and 37.5°C is given as
  $\frac{R_{15}}{R_{37.5}} = \frac{4}{5}$ 

  .

• The resistance at temperature 15°C,
  $R_{15} = R_0 (1 + \alpha \times 15)$ 

  .

• The resistance at temperature 37.5°C,
  $R_{37.5} = R_0 (1 + \alpha \times 37.5)$ 

  .

Step 2: Set up the equation based on the given ratio

$$\frac{R_{15}}{R_{37.5}} = \frac{4}{5}$$

Substituting the expressions for

$R_{15}$

and

$R_{37.5}$

:

$$\frac{R_0 (1 + \alpha \times 15)}{R_0 (1 + \alpha \times 37.5)} = \frac{4}{5}$$

Canceling

$R_0$

from both the numerator and denominator:

$$\frac{1 + 15\alpha}{1 + 37.5\alpha} = \frac{4}{5}$$

Step 3: Solve for $\alpha$

Cross-multiply to solve for

$\alpha$

:

$$5(1 + 15\alpha) = 4(1 + 37.5\alpha)$$

Expanding both sides:

$$5 + 75\alpha = 4 + 150\alpha$$

Simplify:

$$5 - 4 = 150\alpha - 75\alpha$$

$$1 = 75\alpha$$

$$\alpha = \frac{1}{75}$$

Final Answer:

The temperature coefficient of resistance

$\alpha$

is

$\boxed{\frac{1}{75}} \, \text{per °C}$

.

Let's break this down step by step.

1. Resistances connected in parallel

When

$n$

resistors, each of resistance

$r$

, are connected in parallel, the total or equivalent resistance

$R_{\text{parallel}}$

is given by the formula:

$$\frac{1}{R_{\text{parallel}}} = \frac{1}{r} + \frac{1}{r} + \cdots + \frac{1}{r} \quad \text{(n terms)}$$

This simplifies to:

$$\frac{1}{R_{\text{parallel}}} = \frac{n}{r}$$

So, the equivalent resistance is:

$$R_{\text{parallel}} = \frac{r}{n}$$

We're told that the resultant resistance when connected in parallel is

$x$

, so:

$$x = \frac{r}{n}$$

2. Resistances connected in series

When the same

$n$

resistors, each of resistance

$r$

, are connected in series, the total or equivalent resistance

$R_{\text{series}}$

is simply the sum of the individual resistances:

$$R_{\text{series}} = r + r + \cdots + r \quad \text{(n terms)}$$

So:

$$R_{\text{series}} = n \times r$$

3. Relating the two scenarios

From the parallel connection, we know that:

$$x = \frac{r}{n}$$

Thus, the resistance when the resistors are connected in series is:

$$R_{\text{series}} = n \times r = n \times \left( x \times n \right) = r \times n \times x$$

Therefore, the resistance when these

$n$

resistors are connected in series is:

$$\boxed{r \times n \times x}$$