Derive absolute error in equations:
1)Derive absolute error in (e/m)
e/m= 2V/B^2*R^2
2)Derive absolute error in B
B=8uN(Inet)/square root 125a
To derive the absolute error in equations, we need to first determine the partial derivatives of the variables involved with respect to the given equation. Then, we multiply the partial derivatives with their corresponding absolute errors to obtain the absolute error in the desired variable.
1) Derive absolute error in (e/m):
Given equation: e/m = 2V/B^2*R^2
To find the partial derivative of (e/m) with respect to each variable, we follow these steps:
Step 1: Take the partial derivative of (e/m) with respect to V:
∂(e/m)/∂V = 2/B^2*R^2
Step 2: Take the partial derivative of (e/m) with respect to B:
∂(e/m)/∂B = -4V/B^3*R^2
Step 3: Take the partial derivative of (e/m) with respect to R:
∂(e/m)/∂R = -4V/B^2*R
Now, let's multiply each partial derivative with their corresponding absolute errors:
Absolute error in (e/m): Δ(e/m) = (2/B^2*R^2) * ΔV + (-4V/B^3*R^2) * ΔB + (-4V/B^2*R) * ΔR
2) Derive absolute error in B:
Given equation: B = 8uN(Inet)/sqrt(125a)
To find the partial derivative of B with respect to each variable, we follow these steps:
Step 1: Take the partial derivative of B with respect to uN:
∂B/∂uN = 8(Inet)/sqrt(125a)
Step 2: Take the partial derivative of B with respect to Inet:
∂B/∂Inet = 8uN/sqrt(125a)
Step 3: Take the partial derivative of B with respect to a:
∂B/∂a = -4uN(Inet)/sqrt(3125a^3)
Now, let's multiply each partial derivative with their corresponding absolute errors:
Absolute error in B: ΔB = (8(Inet)/sqrt(125a)) * ΔuN + (8uN/sqrt(125a)) * ΔInet + (-4uN(Inet)/sqrt(3125a^3)) * Δa
These are the derived formulas for calculating the absolute error in the given variables in the respective equations.
To derive the absolute error in a given equation, we need to use the concept of partial derivatives. The absolute error (Δx_abs) in a variable x can be calculated using the following formula:
Δx_abs = √((∂x/∂a)² * (Δa)² + (∂x/∂b)² * (Δb)² + ...)
Where (∂x/∂a) represents the partial derivative of x with respect to a, and Δa represents the uncertainty or error in the variable a.
Let's derive the absolute error in each of the given equations:
1) Derive absolute error in (e/m):
The equation is e/m = 2V/(B² * R²)
To find the absolute error in (e/m), we need to consider the uncertainties or errors in V, B, and R.
Δ(e/m)_abs = √((∂(e/m)/∂V)² * (ΔV)² + (∂(e/m)/∂B)² * (ΔB)² + (∂(e/m)/∂R)² * (ΔR)²)
Taking partial derivatives:
(∂(e/m)/∂V) = 2/(B² * R²)
(∂(e/m)/∂B) = -4V/(B³ * R²)
(∂(e/m)/∂R) = -4V/(B² * R³)
Therefore, the absolute error in (e/m) can be calculated as:
Δ(e/m)_abs = √((2/(B² * R²))² * (ΔV)² + (-4V/(B³ * R²))² * (ΔB)² + (-4V/(B² * R³))² * (ΔR)²)
2) Derive absolute error in B:
The equation is B = 8uN(Inet)/√(125a)
To find the absolute error in B, we need to consider the uncertainties or errors in uN (unit Newton), Inet, and a.
ΔB_abs = √((∂B/∂uN)² * (ΔuN)² + (∂B/∂Inet)² * (ΔInet)² + (∂B/∂a)² * (Δa)²)
Taking partial derivatives:
(∂B/∂uN) = 8(Inet)/√(125a)
(∂B/∂Inet) = 8uN/√(125a)
(∂B/∂a) = -8uN(Inet)/(2√(125)a^(3/2))
Therefore, the absolute error in B can be calculated as:
ΔB_abs = √((8(Inet)/√(125a))² * (ΔuN)² + (8uN/√(125a))² * (ΔInet)² + (-8uN(Inet)/(2√(125)a^(3/2)))² * (Δa)²)
These are the general formulas to calculate the absolute error in each variable using partial derivatives.