A 50 ohm resistor consists of a copper portion joined to carbon portion, the resistance of the resistor being the sum of the resistance for 2 portions. Find the resistance of the copper portion of the resistor so that the combined resistance will be constant at all temperature.

To find the resistance of the copper portion of the resistor so that the combined resistance remains constant at all temperatures, we need to consider the temperature coefficients of resistance for both copper and carbon.

The resistance of a material changes with temperature due to its temperature coefficient of resistance (TCR). The TCR is defined as the relative change in resistance per degree Celsius of temperature change. Different materials have different TCR values.

In this case, since the combined resistance remains constant at all temperatures, it means that the change in resistance of the copper portion must be equal and opposite to the change in resistance of the carbon portion as the temperature changes.

To achieve this, we can set up an equation based on the temperature coefficients of resistance for copper and carbon. Let's denote the resistance of the copper portion as Rcopper, the resistance of the carbon portion as Rcarbon, and their respective TCR values as TCRcopper and TCRcarbon.

The equation can be written as:

ΔRcopper = -ΔRcarbon

where ΔRcopper represents the change in resistance of the copper portion and ΔRcarbon represents the change in resistance of the carbon portion.

The change in resistance can be calculated using the formula:

ΔR = R * TCR * ΔT

where ΔT is the change in temperature and R is the resistance.

Now, let's assume the temperature change to be ΔT. Firstly, we need to calculate the change in resistance for the combined resistor. Since the total resistance is 50 ohms and it remains constant, we can write:

ΔRcombined = Rcombined * TCRcombined * ΔT

Since the resistor is composed of two portions in series, the combined resistance can be calculated as:

Rcombined = Rcopper + Rcarbon

Thus, the equation becomes:

ΔRcombined = (Rcopper + Rcarbon) * TCRcombined * ΔT

Since the change in resistance is equal and opposite for the copper and carbon portions, we can write:

ΔRcopper = -ΔRcarbon = -ΔRcombined/2

Substituting this into the equation, we have:

-ΔRcombined/2 = (Rcopper + Rcarbon) * TCRcombined * ΔT

Simplifying, we get:

-(ΔRcombined/2) = (Rcopper + Rcarbon) * TCRcombined * ΔT

Now, we know that the combined resistance of the resistor remains constant, so the change in resistance is zero:

0 = (Rcopper + Rcarbon) * TCRcombined * ΔT

Since ΔT cannot be zero, we can conclude that:

Rcopper + Rcarbon = 0

This means that the resistance of the copper and carbon portions must be equal but with opposite signs. Therefore, to achieve a constant combined resistance, the resistance of the copper portion must be equal to the resistance of the carbon portion, but with a negative sign.

In summary, to maintain constant combined resistance at all temperatures in a 50 ohm resistor consisting of copper and carbon portions, the resistance of the copper portion should be -25 ohms, and the resistance of the carbon portion should be +25 ohms.