91Ó°ÊÓ

Ohm's Law is a fundamental concept in electricity that connects current, voltage, and resistance in an electrical circuit. It is mathematically expressed as \( V = IR \), where \( V \) is the voltage, \( I \) is the current in amperes, and \( R \) is the resistance in ohms. This relationship allows us to determine one of these variables if the other two are known. In the case of an electric space heater, we use a derived formula for power: \( P = IV \). This formula calculates the rate of energy conversion from electrical to thermal energy in the heater, which tells us how much heat is being produced. For example, if a heater has a current of \( 7.25 \mathrm{~A} \) and is operating at \( 120 \mathrm{~V} \), we find the power output, which is the rate of heat supplied:

• Given: \( I = 7.25 \mathrm{~A} \), \( V = 120 \mathrm{~V} \)
• Power, \( P = IV = 7.25 \times 120 = 870 \mathrm{~W} \)

This means the heater supplies energy at a rate of \( 870 \) watts.

The specific heat capacity, often just called specific heat, is an essential property of materials that indicates how much heat energy it takes to change the temperature of a given mass. The formula used is: \[ Q = mc\Delta T \]where:

• \( Q \) is the heat energy absorbed or released (in joules),
• \( m \) is the mass of the substance (in kilograms),
• \( c \) is the specific heat capacity (in joules per kilogram per degree Kelvin), and
• \( \Delta T \) is the change in temperature (in Kelvin or Celsius).

For air, which has a specific heat capacity of around \( 1000 \, \text{J}/(\text{kg}\cdot \text{K}) \), this tells us how efficiently heat energy can raise the air's temperature in a room. Using this, we can analyze how heat from a heater can impact the air: if \( 27 \) kg of air is to be warmed using \( 52200 \) J of energy, specific heat capacity helps determine how much the temperature of the air increases.

Energy conversion is the process of transforming energy from one form to another. In the context of an electric space heater, we are particularly interested in the conversion of electrical energy into thermal energy. When the heater operates with a power output of \( 870 \text{ W} \) (watts), it means \( 870 \, \text{joules} \) of electrical energy are converted into heat every second. To calculate how much energy is supplied over a period of time, we use the formula:

• \( E = P \times t \), where \( t \) is time in seconds.

For example, in 1 minute (\( 60 \) seconds), the energy supplied is:

• Energy, \( E = 870 \, \text{W} \times 60 \, ext{s} = 52200 \, ext{J} \)

This means \( 52200 \, \text{joules} \) of energy are converted to heat, warming the room's air.

Calculating temperature change involves determining how much heat is needed to raise the temperature of a substance by a certain degree. Using the specific heat capacity formula, i.e., \( Q = mc\Delta T \), you can solve for the change in temperature, \( \Delta T \). This is crucial in understanding how efficient the heating process is.In the case of our exercise involving an electric space heater, we want to calculate how much the temperature of air in a room increases. First, you'll need to know:

• The amount of heat energy supplied, \( Q = 52200 \, \text{J} \),
• The mass of the air, \( m = 27 \, \text{kg} \),
• The specific heat capacity of the air, \( c = 1000 \, \text{J/(kg \cdot K)} \).

Plug these values into the formula to solve for \( \Delta T \):

• \( \Delta T = \frac{Q}{mc} = \frac{52200}{27 \times 1000} \approx 1.933 \, \text{K} \)

This shows that the air temperature increases by approximately \( 1.93 \degree C \), which reflects how the heater affects the room.