Question

The rate of change of electric charge with respect to time is called current. Suppose that \(\frac{1}{3} t^{3}+t\) coulombs of charge flow through a wire in \(t\) seconds. Find the current in amperes (coulombs per second) after 3 seconds. When will a 20-ampere fuse in the line blow?

Short answer

The current after 3 seconds is 10 amperes. The fuse will blow at approximately 4.36 seconds.

Step-by-step solution

Define Current as the Derivative

Current is the rate of change of charge with respect to time. To find this, we need to take the derivative of the charge function with respect to time. The charge function is given as \( Q(t) = \frac{1}{3} t^3 + t \). The derivative of \( Q(t) \) with respect to \( t \) is the current \( I(t) \).

Differentiate the Charge Function

Compute the derivative \( I(t) = \frac{d}{dt}(\frac{1}{3} t^3 + t) \). Using the power rule, the derivative of \( \frac{1}{3} t^3 \) is \( t^2 \) and the derivative of \( t \) is \( 1 \). Thus, \( I(t) = t^2 + 1 \).

Evaluate Current at 3 Seconds

Substitute \( t = 3 \) into \( I(t) = t^2 + 1 \). \( I(3) = 3^2 + 1 = 9 + 1 = 10 \) amperes. Thus, the current after 3 seconds is 10 amperes.

Determine When the Fuse Will Blow

To find out when a 20-ampere fuse will blow, set the current \( I(t) = 20 \) and solve for \( t \). This gives \( t^2 + 1 = 20 \). Subtract 1 from both sides to get \( t^2 = 19 \).

Solve for Time \( t \)

Solve the equation \( t^2 = 19 \). Take the square root of both sides, \( t = \pm\sqrt{19} \). Since time cannot be negative, \( t = \sqrt{19} \). Calculate \( \sqrt{19} \) to approximately \( 4.36 \) seconds.

Key concepts

Understanding Derivatives

In calculus, a derivative represents the rate at which a function changes as its input changes. In simpler terms, the derivative shows the slope of the function at any given point.
To find the derivative, you differentiate the function. Differentiation is a process that involves taking the function of interest and applying specific rules to find another function that gives the rate of change.
In the context of our exercise, we have a function for electric charge, which is expressed in terms of time: \( Q(t) = \frac{1}{3}t^3 + t \).
By finding the derivative of this function with respect to time \( t \), we can determine how quickly the charge changes as time progresses. This rate of change is precisely what we define as the current in a circuit.

The Role of Current

Current is an essential concept in electricity and electronics. It represents the flow of electric charge in a circuit.
When we say that current is the rate of change of electric charge, we mean that it tells us how much charge is passing through a point in the circuit per unit of time.
This is usually measured in amperes, where one ampere equals one coulomb of charge passing a certain point per second.

• In the exercise, the current is derived from the charge function \( Q(t) \).
• To find the exact value of the current, we substitute a specific time \( t \) into the derivative \( I(t) \).

This process allows us to calculate the current at any point in time, providing insight into how the circuit behaves.

Applying the Power Rule

The power rule is a fundamental tool in differentiation. It is a straightforward method to find the derivative of functions that have variables raised to any power.
The rule states that if you have a term \( ax^n \), the derivative is \( n \cdot ax^{n-1} \).
Let's apply the power rule to our charge function:

• For \( \frac{1}{3}t^3 \), the derivative is \( t^2 \) because \( 3 \times \frac{1}{3} = 1 \) and reduce the power by one.
• For \( t^1 \), the derivative is 1 as the power reduces to zero and any number to the power of zero is one.

Thus, by applying the power rule, we derive the function for current \( I(t) = t^2 + 1 \), enabling us to find the rate of charge change at any point in time.

Exploring Electric Charge

Electric charge is a fundamental property of matter, associated with how particles interact with electric and magnetic fields. In our exercises, this is the quantity flowing through the wire.
Understanding electric charge is crucial in physics as it explains how electric fields are created and how they affect matter.

• Charge is often measured in coulombs, with one coulomb equaling the charge of approximately \( 6.242 \times 10^{18} \) electrons.
• In circuits, the movement of charge (current) is what powers devices and allows electrical circuits to function.

By observing the behavior of charge over time and employing calculus, we can predict how systems will behave under different conditions. This is vital for designing safe and efficient electrical systems.